id	sid	tid	token	lemma	pos
ejpam-5048	1	1	european	european	PROPN
ejpam-5048	1	2	journal	journal	PROPN
ejpam-5048	1	3	of	of	ADP
ejpam-5048	1	4	pure	pure	ADJ
ejpam-5048	1	5	and	and	CCONJ
ejpam-5048	1	6	applied	apply	VERB
ejpam-5048	1	7	mathematics	mathematic	NOUN
ejpam-5048	1	8	vol	vol	NOUN
ejpam-5048	1	9	.	.	PROPN
ejpam-5048	2	1	17	17	NUM
ejpam-5048	2	2	,	,	PUNCT
ejpam-5048	2	3	no	no	INTJ
ejpam-5048	2	4	.	.	NOUN
ejpam-5048	2	5	1	1	NUM
ejpam-5048	2	6	,	,	PUNCT
ejpam-5048	2	7	2024	2024	NUM
ejpam-5048	2	8	,	,	PUNCT
ejpam-5048	2	9	569	569	NUM
ejpam-5048	2	10	-	-	SYM
ejpam-5048	2	11	581	581	NUM
ejpam-5048	2	12	issn	issn	PROPN
ejpam-5048	2	13	1307	1307	NUM
ejpam-5048	2	14	-	-	SYM
ejpam-5048	2	15	5543	5543	NUM
ejpam-5048	2	16	–	–	PUNCT
ejpam-5048	2	17	ejpam.com	ejpam.com	X
ejpam-5048	2	18	published	publish	VERB
ejpam-5048	2	19	by	by	ADP
ejpam-5048	2	20	new	new	PROPN
ejpam-5048	2	21	york	york	PROPN
ejpam-5048	2	22	business	business	PROPN
ejpam-5048	2	23	global	global	PROPN
ejpam-5048	2	24	on	on	ADP
ejpam-5048	2	25	quasi	quasi	ADJ
ejpam-5048	2	26	generalized	generalized	ADJ
ejpam-5048	2	27	exchange	exchange	NOUN
ejpam-5048	2	28	algebras	algebras	PROPN
ejpam-5048	2	29	young	young	PROPN
ejpam-5048	2	30	bae	bae	PROPN
ejpam-5048	2	31	jun1	jun1	PROPN
ejpam-5048	2	32	,	,	PUNCT
ejpam-5048	2	33	ravikumar	ravikumar	PROPN
ejpam-5048	2	34	bandaru2	bandaru2	PROPN
ejpam-5048	2	35	,	,	PUNCT
ejpam-5048	2	36	rahul	rahul	PROPN
ejpam-5048	2	37	shukla3,∗	shukla3,∗	PROPN
ejpam-5048	2	38	1	1	NUM
ejpam-5048	2	39	department	department	NOUN
ejpam-5048	2	40	of	of	ADP
ejpam-5048	2	41	mathematics	mathematics	PROPN
ejpam-5048	2	42	education	education	NOUN
ejpam-5048	2	43	,	,	PUNCT
ejpam-5048	2	44	gyeongsang	gyeongsang	PROPN
ejpam-5048	2	45	national	national	PROPN
ejpam-5048	2	46	university	university	PROPN
ejpam-5048	2	47	,	,	PUNCT
ejpam-5048	2	48	jinju	jinju	NOUN
ejpam-5048	2	49	52828	52828	NUM
ejpam-5048	2	50	,	,	PUNCT
ejpam-5048	2	51	korea	korea	PROPN
ejpam-5048	2	52	2	2	NUM
ejpam-5048	2	53	department	department	NOUN
ejpam-5048	2	54	of	of	ADP
ejpam-5048	2	55	mathematics	mathematic	NOUN
ejpam-5048	2	56	,	,	PUNCT
ejpam-5048	2	57	school	school	NOUN
ejpam-5048	2	58	of	of	ADP
ejpam-5048	2	59	advanced	advanced	ADJ
ejpam-5048	2	60	sciences	science	NOUN
ejpam-5048	2	61	,	,	PUNCT
ejpam-5048	2	62	vit	vit	PROPN
ejpam-5048	2	63	-	-	PUNCT
ejpam-5048	2	64	ap	ap	PROPN
ejpam-5048	2	65	university	university	PROPN
ejpam-5048	2	66	,	,	PUNCT
ejpam-5048	2	67	andhra	andhra	PROPN
ejpam-5048	2	68	pradesh-522237	pradesh-522237	NOUN
ejpam-5048	2	69	,	,	PUNCT
ejpam-5048	2	70	india	india	PROPN
ejpam-5048	2	71	3	3	NUM
ejpam-5048	2	72	department	department	PROPN
ejpam-5048	2	73	of	of	ADP
ejpam-5048	2	74	mathematical	mathematical	ADJ
ejpam-5048	2	75	sciences	sciences	PROPN
ejpam-5048	2	76	and	and	CCONJ
ejpam-5048	2	77	computing	computing	NOUN
ejpam-5048	2	78	,	,	PUNCT
ejpam-5048	2	79	walter	walter	PROPN
ejpam-5048	2	80	sisulu	sisulu	PROPN
ejpam-5048	2	81	university	university	PROPN
ejpam-5048	2	82	,	,	PUNCT
ejpam-5048	2	83	mthatha	mthatha	NOUN
ejpam-5048	2	84	5117	5117	NUM
ejpam-5048	2	85	,	,	PUNCT
ejpam-5048	2	86	south	south	PROPN
ejpam-5048	2	87	africa	africa	PROPN
ejpam-5048	2	88	abstract	abstract	PROPN
ejpam-5048	2	89	.	.	PUNCT
ejpam-5048	3	1	a	a	DET
ejpam-5048	3	2	new	new	ADJ
ejpam-5048	3	3	type	type	NOUN
ejpam-5048	3	4	of	of	ADP
ejpam-5048	3	5	algebraic	algebraic	ADJ
ejpam-5048	3	6	structure	structure	NOUN
ejpam-5048	3	7	,	,	PUNCT
ejpam-5048	3	8	called	call	VERB
ejpam-5048	3	9	a	a	DET
ejpam-5048	3	10	quasi	quasi	ADJ
ejpam-5048	3	11	generalized	generalized	ADJ
ejpam-5048	3	12	exchange	exchange	NOUN
ejpam-5048	3	13	algebra(qgealgebra	algebra(qgealgebra	PROPN
ejpam-5048	3	14	)	)	PUNCT
ejpam-5048	3	15	,	,	PUNCT
ejpam-5048	3	16	with	with	SCONJ
ejpam-5048	3	17	the	the	DET
ejpam-5048	3	18	ge	ge	PROPN
ejpam-5048	3	19	-	-	PUNCT
ejpam-5048	3	20	algebra	algebra	NOUN
ejpam-5048	3	21	conditions	condition	NOUN
ejpam-5048	3	22	is	be	AUX
ejpam-5048	3	23	introduced	introduce	VERB
ejpam-5048	3	24	and	and	CCONJ
ejpam-5048	3	25	its	its	PRON
ejpam-5048	3	26	properties	property	NOUN
ejpam-5048	3	27	are	be	AUX
ejpam-5048	3	28	investigated	investigate	VERB
ejpam-5048	3	29	.	.	PUNCT
ejpam-5048	4	1	the	the	DET
ejpam-5048	4	2	concepts	concept	NOUN
ejpam-5048	4	3	of	of	ADP
ejpam-5048	4	4	qge	qge	NOUN
ejpam-5048	4	5	-	-	PUNCT
ejpam-5048	4	6	subalgebra	subalgebra	NOUN
ejpam-5048	4	7	,	,	PUNCT
ejpam-5048	4	8	qge	qge	NOUN
ejpam-5048	4	9	-	-	NOUN
ejpam-5048	4	10	filter	filter	NOUN
ejpam-5048	4	11	,	,	PUNCT
ejpam-5048	4	12	closed	close	VERB
ejpam-5048	4	13	qge	qge	NOUN
ejpam-5048	4	14	-	-	NOUN
ejpam-5048	4	15	filter	filter	NOUN
ejpam-5048	4	16	and	and	CCONJ
ejpam-5048	4	17	strong	strong	ADJ
ejpam-5048	4	18	qge	qge	NOUN
ejpam-5048	4	19	-	-	NOUN
ejpam-5048	4	20	filter	filter	NOUN
ejpam-5048	4	21	of	of	ADP
ejpam-5048	4	22	a	a	DET
ejpam-5048	4	23	quasi	quasi	NOUN
ejpam-5048	4	24	gealgebra	gealgebra	NOUN
ejpam-5048	4	25	are	be	AUX
ejpam-5048	4	26	introduced	introduce	VERB
ejpam-5048	4	27	and	and	CCONJ
ejpam-5048	4	28	their	their	PRON
ejpam-5048	4	29	relationships	relationship	NOUN
ejpam-5048	4	30	are	be	AUX
ejpam-5048	4	31	discussed	discuss	VERB
ejpam-5048	4	32	.	.	PUNCT
ejpam-5048	5	1	the	the	DET
ejpam-5048	5	2	conditions	condition	NOUN
ejpam-5048	5	3	for	for	ADP
ejpam-5048	5	4	a	a	DET
ejpam-5048	5	5	subset	subset	NOUN
ejpam-5048	5	6	of	of	ADP
ejpam-5048	5	7	a	a	DET
ejpam-5048	5	8	quasi	quasi	ADJ
ejpam-5048	5	9	ge	ge	PROPN
ejpam-5048	5	10	-	-	PROPN
ejpam-5048	5	11	algebra	algebra	PROPN
ejpam-5048	5	12	to	to	PART
ejpam-5048	5	13	be	be	AUX
ejpam-5048	5	14	a	a	DET
ejpam-5048	5	15	qge	qge	NOUN
ejpam-5048	5	16	-	-	NOUN
ejpam-5048	5	17	filter	filter	NOUN
ejpam-5048	5	18	are	be	AUX
ejpam-5048	5	19	given	give	VERB
ejpam-5048	5	20	.	.	PUNCT
ejpam-5048	6	1	2020	2020	NUM
ejpam-5048	6	2	mathematics	mathematic	NOUN
ejpam-5048	6	3	subject	subject	NOUN
ejpam-5048	6	4	classifications	classification	NOUN
ejpam-5048	6	5	:	:	PUNCT
ejpam-5048	6	6	03g25	03g25	NUM
ejpam-5048	6	7	,	,	PUNCT
ejpam-5048	6	8	06f35	06f35	NUM
ejpam-5048	6	9	key	key	ADJ
ejpam-5048	6	10	words	word	NOUN
ejpam-5048	6	11	and	and	CCONJ
ejpam-5048	6	12	phrases	phrase	NOUN
ejpam-5048	6	13	:	:	PUNCT
ejpam-5048	6	14	quasi	quasi	PROPN
ejpam-5048	6	15	ge	ge	PROPN
ejpam-5048	6	16	-	-	PUNCT
ejpam-5048	6	17	algebra(qge	algebra(qge	PROPN
ejpam-5048	6	18	-	-	PUNCT
ejpam-5048	6	19	algebra	algebra	NOUN
ejpam-5048	6	20	)	)	PUNCT
ejpam-5048	6	21	,	,	PUNCT
ejpam-5048	6	22	qge	qge	NOUN
ejpam-5048	6	23	-	-	PUNCT
ejpam-5048	6	24	subalgebra	subalgebra	NOUN
ejpam-5048	6	25	,	,	PUNCT
ejpam-5048	6	26	(	(	PUNCT
ejpam-5048	6	27	strong	strong	ADJ
ejpam-5048	6	28	,	,	PUNCT
ejpam-5048	6	29	closed	closed	ADJ
ejpam-5048	6	30	)	)	PUNCT
ejpam-5048	6	31	qge	qge	NOUN
ejpam-5048	6	32	-	-	NOUN
ejpam-5048	6	33	filter	filter	NOUN
ejpam-5048	6	34	1	1	NUM
ejpam-5048	6	35	.	.	PUNCT
ejpam-5048	6	36	introduction	introduction	NOUN
ejpam-5048	6	37	l.	l.	PROPN
ejpam-5048	6	38	henkin	henkin	PROPN
ejpam-5048	6	39	and	and	CCONJ
ejpam-5048	6	40	t.	t.	PROPN
ejpam-5048	6	41	skolem	skolem	PROPN
ejpam-5048	6	42	made	make	VERB
ejpam-5048	6	43	significant	significant	ADJ
ejpam-5048	6	44	contributions	contribution	NOUN
ejpam-5048	6	45	to	to	ADP
ejpam-5048	6	46	the	the	DET
ejpam-5048	6	47	field	field	NOUN
ejpam-5048	6	48	of	of	ADP
ejpam-5048	6	49	intuitionistic	intuitionistic	ADJ
ejpam-5048	6	50	and	and	CCONJ
ejpam-5048	6	51	non	non	ADJ
ejpam-5048	6	52	-	-	ADJ
ejpam-5048	6	53	classical	classical	ADJ
ejpam-5048	6	54	logics	logic	NOUN
ejpam-5048	6	55	during	during	ADP
ejpam-5048	6	56	the	the	DET
ejpam-5048	6	57	1950s	1950	NOUN
ejpam-5048	6	58	by	by	ADP
ejpam-5048	6	59	introducing	introduce	VERB
ejpam-5048	6	60	hilbert	hilbert	NOUN
ejpam-5048	6	61	algebras	algebra	NOUN
ejpam-5048	6	62	.	.	PUNCT
ejpam-5048	7	1	an	an	DET
ejpam-5048	7	2	interesting	interesting	ADJ
ejpam-5048	7	3	development	development	NOUN
ejpam-5048	7	4	came	come	VERB
ejpam-5048	7	5	from	from	ADP
ejpam-5048	7	6	a.	a.	PROPN
ejpam-5048	7	7	diego	diego	PROPN
ejpam-5048	7	8	,	,	PUNCT
ejpam-5048	7	9	who	who	PRON
ejpam-5048	7	10	established	establish	VERB
ejpam-5048	7	11	the	the	DET
ejpam-5048	7	12	local	local	ADJ
ejpam-5048	7	13	finiteness	finiteness	NOUN
ejpam-5048	7	14	of	of	ADP
ejpam-5048	7	15	hilbert	hilbert	PROPN
ejpam-5048	7	16	algebras	algebra	NOUN
ejpam-5048	7	17	,	,	PUNCT
ejpam-5048	7	18	as	as	SCONJ
ejpam-5048	7	19	demonstrated	demonstrate	VERB
ejpam-5048	7	20	in	in	ADP
ejpam-5048	7	21	[	[	X
ejpam-5048	7	22	3	3	NUM
ejpam-5048	7	23	]	]	PUNCT
ejpam-5048	7	24	.	.	PUNCT
ejpam-5048	8	1	in	in	ADP
ejpam-5048	8	2	an	an	DET
ejpam-5048	8	3	effort	effort	NOUN
ejpam-5048	8	4	to	to	PART
ejpam-5048	8	5	extend	extend	VERB
ejpam-5048	8	6	the	the	DET
ejpam-5048	8	7	concept	concept	NOUN
ejpam-5048	8	8	of	of	ADP
ejpam-5048	8	9	dual	dual	ADJ
ejpam-5048	8	10	bck	bck	NOUN
ejpam-5048	8	11	-	-	PUNCT
ejpam-5048	8	12	algebras	algebras	PROPN
ejpam-5048	8	13	,	,	PUNCT
ejpam-5048	8	14	h.	h.	PROPN
ejpam-5048	8	15	s.	s.	PROPN
ejpam-5048	8	16	kim	kim	PROPN
ejpam-5048	8	17	and	and	CCONJ
ejpam-5048	8	18	y.	y.	PROPN
ejpam-5048	8	19	h.	h.	PROPN
ejpam-5048	8	20	kim	kim	PROPN
ejpam-5048	8	21	introduced	introduce	VERB
ejpam-5048	8	22	the	the	DET
ejpam-5048	8	23	notion	notion	NOUN
ejpam-5048	8	24	of	of	ADP
ejpam-5048	8	25	be	be	AUX
ejpam-5048	8	26	-	-	PUNCT
ejpam-5048	8	27	algebras	algebra	NOUN
ejpam-5048	8	28	,	,	PUNCT
ejpam-5048	8	29	as	as	SCONJ
ejpam-5048	8	30	discussed	discuss	VERB
ejpam-5048	8	31	in	in	ADP
ejpam-5048	8	32	[	[	X
ejpam-5048	8	33	4	4	NUM
ejpam-5048	8	34	]	]	PUNCT
ejpam-5048	8	35	.	.	PUNCT
ejpam-5048	9	1	drawing	draw	VERB
ejpam-5048	9	2	connections	connection	NOUN
ejpam-5048	9	3	between	between	ADP
ejpam-5048	9	4	hilbert	hilbert	PROPN
ejpam-5048	9	5	algebras	algebras	PROPN
ejpam-5048	9	6	and	and	CCONJ
ejpam-5048	9	7	be	be	AUX
ejpam-5048	9	8	-	-	PUNCT
ejpam-5048	9	9	algebras	algebra	NOUN
ejpam-5048	9	10	,	,	PUNCT
ejpam-5048	9	11	a.	a.	PROPN
ejpam-5048	9	12	rezaei	rezaei	PROPN
ejpam-5048	9	13	et	et	PROPN
ejpam-5048	9	14	al	al	PROPN
ejpam-5048	9	15	.	.	PROPN
ejpam-5048	9	16	explored	explore	VERB
ejpam-5048	9	17	their	their	PRON
ejpam-5048	9	18	interrelations	interrelation	NOUN
ejpam-5048	9	19	,	,	PUNCT
ejpam-5048	9	20	as	as	SCONJ
ejpam-5048	9	21	presented	present	VERB
ejpam-5048	9	22	in	in	ADP
ejpam-5048	9	23	[	[	X
ejpam-5048	9	24	5	5	NUM
ejpam-5048	9	25	]	]	PUNCT
ejpam-5048	9	26	.	.	PUNCT
ejpam-5048	10	1	the	the	DET
ejpam-5048	10	2	process	process	NOUN
ejpam-5048	10	3	of	of	ADP
ejpam-5048	10	4	generalization	generalization	NOUN
ejpam-5048	10	5	is	be	AUX
ejpam-5048	10	6	pivotal	pivotal	ADJ
ejpam-5048	10	7	in	in	ADP
ejpam-5048	10	8	the	the	DET
ejpam-5048	10	9	study	study	NOUN
ejpam-5048	10	10	of	of	ADP
ejpam-5048	10	11	algebraic	algebraic	ADJ
ejpam-5048	10	12	structures	structure	NOUN
ejpam-5048	10	13	,	,	PUNCT
ejpam-5048	10	14	leading	lead	VERB
ejpam-5048	10	15	to	to	ADP
ejpam-5048	10	16	the	the	DET
ejpam-5048	10	17	introduction	introduction	NOUN
ejpam-5048	10	18	of	of	ADP
ejpam-5048	10	19	ge	ge	PROPN
ejpam-5048	10	20	-	-	PUNCT
ejpam-5048	10	21	algebras	algebras	PROPN
ejpam-5048	10	22	by	by	ADP
ejpam-5048	10	23	r.	r.	PROPN
ejpam-5048	10	24	k.	k.	PROPN
ejpam-5048	10	25	bandaru	bandaru	PROPN
ejpam-5048	10	26	et	et	PROPN
ejpam-5048	10	27	al	al	PROPN
ejpam-5048	10	28	.	.	PROPN
ejpam-5048	10	29	,	,	PUNCT
ejpam-5048	10	30	elaborated	elaborate	VERB
ejpam-5048	10	31	in	in	ADP
ejpam-5048	10	32	[	[	X
ejpam-5048	10	33	1	1	NUM
ejpam-5048	10	34	]	]	PUNCT
ejpam-5048	10	35	.	.	PUNCT
ejpam-5048	11	1	an	an	DET
ejpam-5048	11	2	integral	integral	ADJ
ejpam-5048	11	3	facet	facet	NOUN
ejpam-5048	11	4	of	of	ADP
ejpam-5048	11	5	ge	ge	PROPN
ejpam-5048	11	6	-	-	PUNCT
ejpam-5048	11	7	algebras	algebras	PROPN
ejpam-5048	11	8	’	'	PUNCT
ejpam-5048	11	9	advancement	advancement	NOUN
ejpam-5048	11	10	lies	lie	VERB
ejpam-5048	11	11	in	in	ADP
ejpam-5048	11	12	filter	filter	NOUN
ejpam-5048	11	13	theory	theory	NOUN
ejpam-5048	11	14	,	,	PUNCT
ejpam-5048	11	15	which	which	PRON
ejpam-5048	11	16	was	be	AUX
ejpam-5048	11	17	leveraged	leverage	VERB
ejpam-5048	11	18	by	by	ADP
ejpam-5048	11	19	r.	r.	PROPN
ejpam-5048	11	20	k.	k.	PROPN
ejpam-5048	11	21	bandaru	bandaru	PROPN
ejpam-5048	11	22	et	et	PROPN
ejpam-5048	11	23	al	al	PROPN
ejpam-5048	11	24	.	.	PROPN
ejpam-5048	12	1	in	in	ADP
ejpam-5048	12	2	the	the	DET
ejpam-5048	12	3	establishment	establishment	NOUN
ejpam-5048	12	4	of	of	ADP
ejpam-5048	12	5	belligerent	belligerent	ADJ
ejpam-5048	12	6	ge	ge	NOUN
ejpam-5048	12	7	-	-	PUNCT
ejpam-5048	12	8	filters	filter	NOUN
ejpam-5048	12	9	within	within	ADP
ejpam-5048	12	10	ge	ge	PROPN
ejpam-5048	12	11	-	-	PUNCT
ejpam-5048	12	12	algebras	algebras	PROPN
ejpam-5048	12	13	.	.	PUNCT
ejpam-5048	13	1	their	their	PRON
ejpam-5048	13	2	properties	property	NOUN
ejpam-5048	13	3	were	be	AUX
ejpam-5048	13	4	thoroughly	thoroughly	ADV
ejpam-5048	13	5	investigated	investigate	VERB
ejpam-5048	13	6	,	,	PUNCT
ejpam-5048	13	7	as	as	SCONJ
ejpam-5048	13	8	documented	document	VERB
ejpam-5048	13	9	in	in	ADP
ejpam-5048	13	10	[	[	X
ejpam-5048	13	11	2	2	NUM
ejpam-5048	13	12	]	]	PUNCT
ejpam-5048	13	13	.	.	PUNCT
ejpam-5048	14	1	∗corresponding	∗corresponde	VERB
ejpam-5048	14	2	author	author	NOUN
ejpam-5048	14	3	.	.	PUNCT
ejpam-5048	15	1	doi	doi	NOUN
ejpam-5048	15	2	:	:	PUNCT
ejpam-5048	15	3	https://doi.org/10.29020/nybg.ejpam.v17i1.5048	https://doi.org/10.29020/nybg.ejpam.v17i1.5048	ADJ
ejpam-5048	15	4	email	email	NOUN
ejpam-5048	15	5	addresses	address	NOUN
ejpam-5048	15	6	:	:	PUNCT
ejpam-5048	15	7	skywine@gmail.com	skywine@gmail.com	X
ejpam-5048	15	8	(	(	PUNCT
ejpam-5048	15	9	y.	y.	PROPN
ejpam-5048	15	10	b.	b.	PROPN
ejpam-5048	15	11	jun	jun	PROPN
ejpam-5048	15	12	)	)	PUNCT
ejpam-5048	15	13	,	,	PUNCT
ejpam-5048	15	14	ravimaths83@gmail.com	ravimaths83@gmail.com	PROPN
ejpam-5048	15	15	(	(	PUNCT
ejpam-5048	15	16	r.	r.	PROPN
ejpam-5048	15	17	bandaru	bandaru	PROPN
ejpam-5048	15	18	)	)	PUNCT
ejpam-5048	15	19	,	,	PUNCT
ejpam-5048	15	20	rshukla@wsu.ac.za	rshukla@wsu.ac.za	NOUN
ejpam-5048	15	21	(	(	PUNCT
ejpam-5048	15	22	r.	r.	NOUN
ejpam-5048	15	23	shukla	shukla	PROPN
ejpam-5048	15	24	)	)	PUNCT
ejpam-5048	15	25	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5048	16	1	569	569	NUM
ejpam-5048	16	2	©	©	ADP
ejpam-5048	16	3	2024	2024	NUM
ejpam-5048	16	4	ejpam	ejpam	NOUN
ejpam-5048	16	5	all	all	DET
ejpam-5048	16	6	rights	right	NOUN
ejpam-5048	16	7	reserved	reserve	VERB
ejpam-5048	16	8	.	.	PUNCT
ejpam-5048	17	1	y.	y.	PROPN
ejpam-5048	17	2	b.	b.	PROPN
ejpam-5048	17	3	jun	jun	PROPN
ejpam-5048	17	4	,	,	PUNCT
ejpam-5048	17	5	ravikumar	ravikumar	PROPN
ejpam-5048	17	6	bandaru	bandaru	PROPN
ejpam-5048	17	7	,	,	PUNCT
ejpam-5048	17	8	rahul	rahul	PROPN
ejpam-5048	17	9	shukla	shukla	PROPN
ejpam-5048	17	10	/	/	SYM
ejpam-5048	17	11	eur	eur	PROPN
ejpam-5048	17	12	.	.	PUNCT
ejpam-5048	18	1	j.	j.	PROPN
ejpam-5048	18	2	pure	pure	PROPN
ejpam-5048	18	3	appl	appl	PROPN
ejpam-5048	18	4	.	.	PROPN
ejpam-5048	18	5	math	math	PROPN
ejpam-5048	18	6	,	,	PUNCT
ejpam-5048	18	7	17	17	NUM
ejpam-5048	18	8	(	(	PUNCT
ejpam-5048	18	9	1	1	NUM
ejpam-5048	18	10	)	)	PUNCT
ejpam-5048	18	11	(	(	PUNCT
ejpam-5048	18	12	2024	2024	NUM
ejpam-5048	18	13	)	)	PUNCT
ejpam-5048	18	14	,	,	PUNCT
ejpam-5048	18	15	569	569	NUM
ejpam-5048	18	16	-	-	SYM
ejpam-5048	18	17	581	581	NUM
ejpam-5048	18	18	570	570	NUM
ejpam-5048	18	19	in	in	ADP
ejpam-5048	18	20	this	this	DET
ejpam-5048	18	21	paper	paper	NOUN
ejpam-5048	18	22	,	,	PUNCT
ejpam-5048	18	23	we	we	PRON
ejpam-5048	18	24	introduce	introduce	VERB
ejpam-5048	18	25	a	a	DET
ejpam-5048	18	26	new	new	ADJ
ejpam-5048	18	27	type	type	NOUN
ejpam-5048	18	28	of	of	ADP
ejpam-5048	18	29	algebraic	algebraic	ADJ
ejpam-5048	18	30	structure	structure	NOUN
ejpam-5048	18	31	,	,	PUNCT
ejpam-5048	18	32	called	call	VERB
ejpam-5048	18	33	a	a	DET
ejpam-5048	18	34	quasi	quasi	ADJ
ejpam-5048	18	35	ge	ge	PROPN
ejpam-5048	18	36	-	-	PROPN
ejpam-5048	18	37	algebra	algebra	PROPN
ejpam-5048	18	38	(	(	PUNCT
ejpam-5048	18	39	briefly	briefly	ADV
ejpam-5048	18	40	,	,	PUNCT
ejpam-5048	18	41	qge	qge	NOUN
ejpam-5048	18	42	-	-	NOUN
ejpam-5048	18	43	algebra	algebra	NOUN
ejpam-5048	18	44	)	)	PUNCT
ejpam-5048	18	45	,	,	PUNCT
ejpam-5048	18	46	with	with	ADP
ejpam-5048	18	47	the	the	DET
ejpam-5048	18	48	conditions	condition	NOUN
ejpam-5048	18	49	of	of	ADP
ejpam-5048	18	50	ge	ge	PROPN
ejpam-5048	18	51	-	-	PUNCT
ejpam-5048	18	52	algebra	algebra	PROPN
ejpam-5048	18	53	and	and	CCONJ
ejpam-5048	18	54	investigate	investigate	VERB
ejpam-5048	18	55	its	its	PRON
ejpam-5048	18	56	properties	property	NOUN
ejpam-5048	18	57	.	.	PUNCT
ejpam-5048	19	1	we	we	PRON
ejpam-5048	19	2	show	show	VERB
ejpam-5048	19	3	that	that	SCONJ
ejpam-5048	19	4	ge	ge	PROPN
ejpam-5048	19	5	-	-	PUNCT
ejpam-5048	19	6	algebra	algebra	PROPN
ejpam-5048	19	7	and	and	CCONJ
ejpam-5048	19	8	qge	qge	NOUN
ejpam-5048	19	9	-	-	PUNCT
ejpam-5048	19	10	algebra	algebra	NOUN
ejpam-5048	19	11	are	be	AUX
ejpam-5048	19	12	independent	independent	ADJ
ejpam-5048	19	13	of	of	ADP
ejpam-5048	19	14	each	each	DET
ejpam-5048	19	15	other	other	ADJ
ejpam-5048	19	16	through	through	ADP
ejpam-5048	19	17	examples	example	NOUN
ejpam-5048	19	18	.	.	PUNCT
ejpam-5048	20	1	we	we	PRON
ejpam-5048	20	2	introduce	introduce	VERB
ejpam-5048	20	3	the	the	DET
ejpam-5048	20	4	substructure	substructure	NOUN
ejpam-5048	20	5	of	of	ADP
ejpam-5048	20	6	quasi	quasi	PROPN
ejpam-5048	20	7	ge	ge	PROPN
ejpam-5048	20	8	-	-	PROPN
ejpam-5048	20	9	algebra	algebra	PROPN
ejpam-5048	20	10	called	call	VERB
ejpam-5048	20	11	qge	qge	NOUN
ejpam-5048	20	12	-	-	PUNCT
ejpam-5048	20	13	subalgebra	subalgebra	NOUN
ejpam-5048	20	14	,	,	PUNCT
ejpam-5048	20	15	qgefilter	qgefilter	NOUN
ejpam-5048	20	16	,	,	PUNCT
ejpam-5048	20	17	strong	strong	ADJ
ejpam-5048	20	18	qge	qge	NOUN
ejpam-5048	20	19	-	-	NOUN
ejpam-5048	20	20	filter	filter	NOUN
ejpam-5048	20	21	,	,	PUNCT
ejpam-5048	20	22	and	and	CCONJ
ejpam-5048	20	23	closed	close	VERB
ejpam-5048	20	24	qge	qge	NOUN
ejpam-5048	20	25	-	-	NOUN
ejpam-5048	20	26	filter	filter	NOUN
ejpam-5048	20	27	,	,	PUNCT
ejpam-5048	20	28	and	and	CCONJ
ejpam-5048	20	29	further	far	ADV
ejpam-5048	20	30	explore	explore	VERB
ejpam-5048	20	31	the	the	DET
ejpam-5048	20	32	relevant	relevant	ADJ
ejpam-5048	20	33	properties	property	NOUN
ejpam-5048	20	34	and	and	CCONJ
ejpam-5048	20	35	interrelationship	interrelationship	NOUN
ejpam-5048	20	36	.	.	PUNCT
ejpam-5048	21	1	we	we	PRON
ejpam-5048	21	2	provide	provide	VERB
ejpam-5048	21	3	several	several	ADJ
ejpam-5048	21	4	conditions	condition	NOUN
ejpam-5048	21	5	for	for	ADP
ejpam-5048	21	6	a	a	DET
ejpam-5048	21	7	subset	subset	NOUN
ejpam-5048	21	8	of	of	ADP
ejpam-5048	21	9	a	a	DET
ejpam-5048	21	10	qge	qge	NOUN
ejpam-5048	21	11	-	-	NOUN
ejpam-5048	21	12	algebra	algebra	NOUN
ejpam-5048	21	13	to	to	PART
ejpam-5048	21	14	be	be	AUX
ejpam-5048	21	15	a	a	DET
ejpam-5048	21	16	qge	qge	NOUN
ejpam-5048	21	17	-	-	NOUN
ejpam-5048	21	18	filter	filter	NOUN
ejpam-5048	21	19	.	.	PUNCT
ejpam-5048	22	1	2	2	X
ejpam-5048	22	2	.	.	X
ejpam-5048	22	3	preliminaries	preliminary	NOUN
ejpam-5048	22	4	we	we	PRON
ejpam-5048	22	5	display	display	VERB
ejpam-5048	22	6	the	the	DET
ejpam-5048	22	7	basic	basic	ADJ
ejpam-5048	22	8	notions	notion	NOUN
ejpam-5048	22	9	on	on	ADP
ejpam-5048	22	10	ge	ge	PROPN
ejpam-5048	22	11	-	-	PUNCT
ejpam-5048	22	12	algebras	algebras	PROPN
ejpam-5048	22	13	.	.	PUNCT
ejpam-5048	23	1	a	a	DET
ejpam-5048	23	2	ge	ge	PROPN
ejpam-5048	23	3	-	-	PUNCT
ejpam-5048	23	4	algebra	algebra	PROPN
ejpam-5048	23	5	(	(	PUNCT
ejpam-5048	23	6	see	see	VERB
ejpam-5048	23	7	[	[	X
ejpam-5048	23	8	[	[	X
ejpam-5048	23	9	1	1	NUM
ejpam-5048	23	10	]	]	NOUN
ejpam-5048	23	11	]	]	PUNCT
ejpam-5048	23	12	)	)	PUNCT
ejpam-5048	23	13	is	be	AUX
ejpam-5048	23	14	a	a	DET
ejpam-5048	23	15	non	non	ADJ
ejpam-5048	23	16	-	-	ADJ
ejpam-5048	23	17	empty	empty	ADJ
ejpam-5048	23	18	set	set	NOUN
ejpam-5048	23	19	x	x	PUNCT
ejpam-5048	23	20	with	with	ADP
ejpam-5048	23	21	a	a	DET
ejpam-5048	23	22	constant	constant	ADJ
ejpam-5048	23	23	1	1	NUM
ejpam-5048	23	24	and	and	CCONJ
ejpam-5048	23	25	a	a	DET
ejpam-5048	23	26	binary	binary	ADJ
ejpam-5048	23	27	operation	operation	NOUN
ejpam-5048	23	28	“	"	PUNCT
ejpam-5048	23	29	∗	∗	NOUN
ejpam-5048	23	30	”	"	PUNCT
ejpam-5048	23	31	satisfying	satisfy	VERB
ejpam-5048	23	32	the	the	DET
ejpam-5048	23	33	following	follow	VERB
ejpam-5048	23	34	axioms	axiom	NOUN
ejpam-5048	23	35	:	:	PUNCT
ejpam-5048	23	36	(	(	PUNCT
ejpam-5048	23	37	ge1	ge1	NOUN
ejpam-5048	23	38	)	)	PUNCT
ejpam-5048	23	39	ϖ	ϖ	NOUN
ejpam-5048	23	40	∗ϖ	∗ϖ	NOUN
ejpam-5048	23	41	=	=	SYM
ejpam-5048	23	42	1	1	NUM
ejpam-5048	23	43	,	,	PUNCT
ejpam-5048	23	44	(	(	PUNCT
ejpam-5048	23	45	ge2	ge2	NOUN
ejpam-5048	23	46	)	)	PUNCT
ejpam-5048	23	47	1	1	NUM
ejpam-5048	23	48	∗ϖ	∗ϖ	NOUN
ejpam-5048	23	49	=	=	SYM
ejpam-5048	23	50	ϖ	ϖ	NOUN
ejpam-5048	23	51	,	,	PUNCT
ejpam-5048	23	52	(	(	PUNCT
ejpam-5048	23	53	ge3	ge3	NOUN
ejpam-5048	23	54	)	)	PUNCT
ejpam-5048	23	55	ϖ	ϖ	NOUN
ejpam-5048	23	56	∗	∗	NOUN
ejpam-5048	23	57	(	(	PUNCT
ejpam-5048	23	58	π	π	PROPN
ejpam-5048	23	59	∗	∗	PROPN
ejpam-5048	23	60	η	η	PROPN
ejpam-5048	23	61	)	)	PUNCT
ejpam-5048	23	62	=	=	SYM
ejpam-5048	23	63	ϖ	ϖ	NOUN
ejpam-5048	23	64	∗	∗	NOUN
ejpam-5048	23	65	(	(	PUNCT
ejpam-5048	23	66	π	π	NOUN
ejpam-5048	23	67	∗	∗	NOUN
ejpam-5048	23	68	(	(	PUNCT
ejpam-5048	23	69	ϖ	ϖ	X
ejpam-5048	23	70	∗	∗	X
ejpam-5048	23	71	η	η	PROPN
ejpam-5048	23	72	)	)	PUNCT
ejpam-5048	23	73	)	)	PUNCT
ejpam-5048	23	74	for	for	ADP
ejpam-5048	23	75	all	all	DET
ejpam-5048	23	76	ϖ,π	ϖ,π	PROPN
ejpam-5048	23	77	,	,	PUNCT
ejpam-5048	23	78	η	η	PROPN
ejpam-5048	23	79	∈	∈	PROPN
ejpam-5048	23	80	x.	x.	NOUN
ejpam-5048	23	81	in	in	ADP
ejpam-5048	23	82	a	a	DET
ejpam-5048	23	83	ge	ge	PROPN
ejpam-5048	23	84	-	-	PUNCT
ejpam-5048	23	85	algebra	algebra	PROPN
ejpam-5048	23	86	x	x	NOUN
ejpam-5048	23	87	,	,	PUNCT
ejpam-5048	23	88	a	a	DET
ejpam-5048	23	89	binary	binary	ADJ
ejpam-5048	23	90	relation	relation	NOUN
ejpam-5048	23	91	“	"	PUNCT
ejpam-5048	23	92	≤	≤	NUM
ejpam-5048	23	93	”	"	PUNCT
ejpam-5048	23	94	is	be	AUX
ejpam-5048	23	95	defined	define	VERB
ejpam-5048	23	96	by	by	ADP
ejpam-5048	23	97	(	(	PUNCT
ejpam-5048	23	98	∀ϖ,π	∀ϖ,π	SYM
ejpam-5048	23	99	∈	∈	PROPN
ejpam-5048	23	100	x	x	X
ejpam-5048	23	101	)	)	PUNCT
ejpam-5048	23	102	(	(	PUNCT
ejpam-5048	23	103	ϖ	ϖ	NOUN
ejpam-5048	23	104	≤	≤	NUM
ejpam-5048	23	105	π	π	PROPN
ejpam-5048	23	106	⇔	⇔	X
ejpam-5048	23	107	ϖ	ϖ	PROPN
ejpam-5048	23	108	∗	∗	X
ejpam-5048	23	109	π	π	X
ejpam-5048	23	110	=	=	PUNCT
ejpam-5048	23	111	1	1	NUM
ejpam-5048	23	112	)	)	PUNCT
ejpam-5048	23	113	.	.	PUNCT
ejpam-5048	24	1	(	(	PUNCT
ejpam-5048	24	2	1	1	X
ejpam-5048	24	3	)	)	PUNCT
ejpam-5048	24	4	every	every	DET
ejpam-5048	24	5	ge	ge	PROPN
ejpam-5048	24	6	-	-	PUNCT
ejpam-5048	24	7	algebra	algebra	PROPN
ejpam-5048	24	8	x	x	PRON
ejpam-5048	24	9	satisfies	satisfy	VERB
ejpam-5048	24	10	the	the	DET
ejpam-5048	24	11	following	follow	VERB
ejpam-5048	24	12	items	item	NOUN
ejpam-5048	24	13	(	(	PUNCT
ejpam-5048	24	14	see	see	VERB
ejpam-5048	24	15	[	[	X
ejpam-5048	24	16	[	[	X
ejpam-5048	24	17	1	1	NUM
ejpam-5048	24	18	]	]	NOUN
ejpam-5048	24	19	]	]	PUNCT
ejpam-5048	24	20	)	)	PUNCT
ejpam-5048	24	21	.	.	PUNCT
ejpam-5048	25	1	(	(	PUNCT
ejpam-5048	25	2	∀ϖ	∀ϖ	ADJ
ejpam-5048	25	3	∈	∈	NOUN
ejpam-5048	25	4	x	x	NOUN
ejpam-5048	25	5	)	)	PUNCT
ejpam-5048	25	6	(	(	PUNCT
ejpam-5048	25	7	ϖ	ϖ	NOUN
ejpam-5048	25	8	∗	∗	NOUN
ejpam-5048	25	9	1	1	NUM
ejpam-5048	25	10	=	=	SYM
ejpam-5048	25	11	1	1	NUM
ejpam-5048	25	12	)	)	PUNCT
ejpam-5048	25	13	.	.	PUNCT
ejpam-5048	26	1	(	(	PUNCT
ejpam-5048	26	2	2	2	X
ejpam-5048	26	3	)	)	PUNCT
ejpam-5048	26	4	(	(	PUNCT
ejpam-5048	26	5	∀ϖ,π	∀ϖ,π	SYM
ejpam-5048	26	6	∈	∈	PROPN
ejpam-5048	26	7	x	x	X
ejpam-5048	26	8	)	)	PUNCT
ejpam-5048	26	9	(	(	PUNCT
ejpam-5048	26	10	ϖ	ϖ	NOUN
ejpam-5048	26	11	∗	∗	NOUN
ejpam-5048	26	12	(	(	PUNCT
ejpam-5048	26	13	ϖ	ϖ	X
ejpam-5048	26	14	∗	∗	X
ejpam-5048	26	15	π	π	NOUN
ejpam-5048	26	16	)	)	PUNCT
ejpam-5048	26	17	=	=	SYM
ejpam-5048	26	18	ϖ	ϖ	PROPN
ejpam-5048	26	19	∗	∗	X
ejpam-5048	26	20	π	π	PROPN
ejpam-5048	26	21	)	)	PUNCT
ejpam-5048	26	22	.	.	PUNCT
ejpam-5048	27	1	(	(	PUNCT
ejpam-5048	27	2	3	3	X
ejpam-5048	27	3	)	)	PUNCT
ejpam-5048	27	4	(	(	PUNCT
ejpam-5048	27	5	∀ϖ,π	∀ϖ,π	SYM
ejpam-5048	27	6	∈	∈	PROPN
ejpam-5048	27	7	x	x	X
ejpam-5048	27	8	)	)	PUNCT
ejpam-5048	27	9	(	(	PUNCT
ejpam-5048	27	10	ϖ	ϖ	NOUN
ejpam-5048	27	11	≤	≤	NUM
ejpam-5048	27	12	π	π	PROPN
ejpam-5048	27	13	∗ϖ	∗ϖ	PROPN
ejpam-5048	27	14	)	)	PUNCT
ejpam-5048	27	15	.	.	PUNCT
ejpam-5048	28	1	(	(	PUNCT
ejpam-5048	28	2	4	4	NUM
ejpam-5048	28	3	)	)	PUNCT
ejpam-5048	28	4	(	(	PUNCT
ejpam-5048	28	5	∀ϖ,π	∀ϖ,π	X
ejpam-5048	28	6	,	,	PUNCT
ejpam-5048	28	7	η	η	PROPN
ejpam-5048	28	8	∈	∈	PROPN
ejpam-5048	28	9	x	x	X
ejpam-5048	28	10	)	)	PUNCT
ejpam-5048	28	11	(	(	PUNCT
ejpam-5048	28	12	ϖ	ϖ	NOUN
ejpam-5048	28	13	∗	∗	NOUN
ejpam-5048	28	14	(	(	PUNCT
ejpam-5048	28	15	π	π	PROPN
ejpam-5048	28	16	∗	∗	X
ejpam-5048	28	17	η	η	PROPN
ejpam-5048	28	18	)	)	PUNCT
ejpam-5048	28	19	≤	≤	PROPN
ejpam-5048	29	1	π	π	PROPN
ejpam-5048	29	2	∗	∗	NOUN
ejpam-5048	29	3	(	(	PUNCT
ejpam-5048	29	4	ϖ	ϖ	X
ejpam-5048	29	5	∗	∗	X
ejpam-5048	29	6	η	η	PROPN
ejpam-5048	29	7	)	)	PUNCT
ejpam-5048	29	8	)	)	PUNCT
ejpam-5048	29	9	.	.	PUNCT
ejpam-5048	30	1	(	(	PUNCT
ejpam-5048	30	2	5	5	NUM
ejpam-5048	30	3	)	)	PUNCT
ejpam-5048	30	4	(	(	PUNCT
ejpam-5048	30	5	∀ϖ	∀ϖ	NOUN
ejpam-5048	30	6	∈	∈	NOUN
ejpam-5048	30	7	x	x	NOUN
ejpam-5048	30	8	)	)	PUNCT
ejpam-5048	30	9	(	(	PUNCT
ejpam-5048	30	10	1	1	NUM
ejpam-5048	30	11	≤	≤	NUM
ejpam-5048	30	12	ϖ	ϖ	NOUN
ejpam-5048	30	13	⇒	⇒	NOUN
ejpam-5048	30	14	ϖ	ϖ	X
ejpam-5048	30	15	=	=	NOUN
ejpam-5048	30	16	1	1	NUM
ejpam-5048	30	17	)	)	PUNCT
ejpam-5048	30	18	.	.	PUNCT
ejpam-5048	31	1	(	(	PUNCT
ejpam-5048	31	2	6	6	NUM
ejpam-5048	31	3	)	)	PUNCT
ejpam-5048	31	4	(	(	PUNCT
ejpam-5048	31	5	∀ϖ,π	∀ϖ,π	SYM
ejpam-5048	31	6	∈	∈	PROPN
ejpam-5048	31	7	x	x	X
ejpam-5048	31	8	)	)	PUNCT
ejpam-5048	31	9	(	(	PUNCT
ejpam-5048	31	10	ϖ	ϖ	NOUN
ejpam-5048	31	11	≤	≤	NUM
ejpam-5048	31	12	(	(	PUNCT
ejpam-5048	31	13	π	π	NOUN
ejpam-5048	31	14	∗ϖ	∗ϖ	PROPN
ejpam-5048	31	15	)	)	PUNCT
ejpam-5048	31	16	∗ϖ	∗ϖ	PROPN
ejpam-5048	31	17	)	)	PUNCT
ejpam-5048	31	18	.	.	PUNCT
ejpam-5048	32	1	(	(	PUNCT
ejpam-5048	32	2	7	7	X
ejpam-5048	32	3	)	)	PUNCT
ejpam-5048	32	4	(	(	PUNCT
ejpam-5048	32	5	∀ϖ,π	∀ϖ,π	SYM
ejpam-5048	32	6	∈	∈	PROPN
ejpam-5048	32	7	x	x	X
ejpam-5048	32	8	)	)	PUNCT
ejpam-5048	32	9	(	(	PUNCT
ejpam-5048	32	10	ϖ	ϖ	NOUN
ejpam-5048	32	11	≤	≤	NUM
ejpam-5048	32	12	(	(	PUNCT
ejpam-5048	32	13	ϖ	ϖ	NOUN
ejpam-5048	32	14	∗	∗	X
ejpam-5048	32	15	π	π	NOUN
ejpam-5048	32	16	)	)	PUNCT
ejpam-5048	32	17	∗	∗	PROPN
ejpam-5048	32	18	π	π	PROPN
ejpam-5048	32	19	)	)	PUNCT
ejpam-5048	32	20	.	.	PUNCT
ejpam-5048	33	1	(	(	PUNCT
ejpam-5048	33	2	8)	8)	NUM
ejpam-5048	33	3	(	(	PUNCT
ejpam-5048	33	4	∀ϖ,π	∀ϖ,π	PROPN
ejpam-5048	33	5	,	,	PUNCT
ejpam-5048	33	6	η	η	PROPN
ejpam-5048	33	7	∈	∈	PROPN
ejpam-5048	33	8	x	x	X
ejpam-5048	33	9	)	)	PUNCT
ejpam-5048	33	10	(	(	PUNCT
ejpam-5048	33	11	ϖ	ϖ	NOUN
ejpam-5048	33	12	≤	≤	NUM
ejpam-5048	33	13	π	π	PROPN
ejpam-5048	33	14	∗	∗	PROPN
ejpam-5048	33	15	η	η	PROPN
ejpam-5048	33	16	⇔	⇔	PROPN
ejpam-5048	33	17	π	π	PROPN
ejpam-5048	33	18	≤	≤	PROPN
ejpam-5048	33	19	ϖ	ϖ	PROPN
ejpam-5048	33	20	∗	∗	X
ejpam-5048	33	21	η	η	PROPN
ejpam-5048	33	22	)	)	PUNCT
ejpam-5048	33	23	.	.	PUNCT
ejpam-5048	34	1	(	(	PUNCT
ejpam-5048	34	2	9	9	X
ejpam-5048	34	3	)	)	SYM
ejpam-5048	34	4	3	3	NUM
ejpam-5048	34	5	.	.	PUNCT
ejpam-5048	34	6	quasi	quasi	PROPN
ejpam-5048	34	7	ge	ge	PROPN
ejpam-5048	34	8	-	-	PUNCT
ejpam-5048	34	9	algebras	algebras	PROPN
ejpam-5048	34	10	in	in	ADP
ejpam-5048	34	11	a	a	DET
ejpam-5048	34	12	ge	ge	PROPN
ejpam-5048	34	13	-	-	PUNCT
ejpam-5048	34	14	algebra	algebra	PROPN
ejpam-5048	34	15	x	x	NOUN
ejpam-5048	34	16	,	,	PUNCT
ejpam-5048	34	17	we	we	PRON
ejpam-5048	34	18	consider	consider	VERB
ejpam-5048	34	19	the	the	DET
ejpam-5048	34	20	following	follow	VERB
ejpam-5048	34	21	equality	equality	NOUN
ejpam-5048	34	22	:	:	PUNCT
ejpam-5048	34	23	(	(	PUNCT
ejpam-5048	34	24	∀κ	∀κ	ADV
ejpam-5048	34	25	,	,	PUNCT
ejpam-5048	34	26	δ	δ	PROPN
ejpam-5048	34	27	,	,	PUNCT
ejpam-5048	34	28	ς	ς	PROPN
ejpam-5048	34	29	∈	∈	PROPN
ejpam-5048	34	30	x)(κ	x)(κ	PROPN
ejpam-5048	34	31	∗	∗	NOUN
ejpam-5048	34	32	δ	δ	PROPN
ejpam-5048	34	33	=	=	PRON
ejpam-5048	34	34	(	(	PUNCT
ejpam-5048	34	35	ς	ς	PROPN
ejpam-5048	34	36	∗	∗	NOUN
ejpam-5048	34	37	κ	κ	NOUN
ejpam-5048	34	38	)	)	PUNCT
ejpam-5048	34	39	∗	∗	NOUN
ejpam-5048	34	40	(	(	PUNCT
ejpam-5048	34	41	ς	ς	PROPN
ejpam-5048	34	42	∗	∗	X
ejpam-5048	34	43	δ	δ	PROPN
ejpam-5048	34	44	)	)	PUNCT
ejpam-5048	34	45	)	)	PUNCT
ejpam-5048	34	46	.	.	PUNCT
ejpam-5048	35	1	(	(	PUNCT
ejpam-5048	35	2	10	10	NUM
ejpam-5048	35	3	)	)	PUNCT
ejpam-5048	35	4	the	the	DET
ejpam-5048	35	5	following	follow	VERB
ejpam-5048	35	6	example	example	NOUN
ejpam-5048	35	7	shows	show	VERB
ejpam-5048	35	8	that	that	SCONJ
ejpam-5048	35	9	a	a	DET
ejpam-5048	35	10	ge	ge	PROPN
ejpam-5048	35	11	-	-	PUNCT
ejpam-5048	35	12	algebra	algebra	PROPN
ejpam-5048	35	13	may	may	AUX
ejpam-5048	35	14	not	not	PART
ejpam-5048	35	15	satisfy	satisfy	VERB
ejpam-5048	35	16	the	the	DET
ejpam-5048	35	17	condition	condition	NOUN
ejpam-5048	35	18	(	(	PUNCT
ejpam-5048	35	19	10	10	NUM
ejpam-5048	35	20	)	)	PUNCT
ejpam-5048	35	21	.	.	PUNCT
ejpam-5048	36	1	y.	y.	PROPN
ejpam-5048	36	2	b.	b.	PROPN
ejpam-5048	36	3	jun	jun	PROPN
ejpam-5048	36	4	,	,	PUNCT
ejpam-5048	36	5	ravikumar	ravikumar	PROPN
ejpam-5048	36	6	bandaru	bandaru	PROPN
ejpam-5048	36	7	,	,	PUNCT
ejpam-5048	36	8	rahul	rahul	PROPN
ejpam-5048	36	9	shukla	shukla	PROPN
ejpam-5048	36	10	/	/	SYM
ejpam-5048	36	11	eur	eur	PROPN
ejpam-5048	36	12	.	.	PUNCT
ejpam-5048	37	1	j.	j.	PROPN
ejpam-5048	37	2	pure	pure	PROPN
ejpam-5048	37	3	appl	appl	PROPN
ejpam-5048	37	4	.	.	PROPN
ejpam-5048	37	5	math	math	PROPN
ejpam-5048	37	6	,	,	PUNCT
ejpam-5048	37	7	17	17	NUM
ejpam-5048	37	8	(	(	PUNCT
ejpam-5048	37	9	1	1	NUM
ejpam-5048	37	10	)	)	PUNCT
ejpam-5048	37	11	(	(	PUNCT
ejpam-5048	37	12	2024	2024	NUM
ejpam-5048	37	13	)	)	PUNCT
ejpam-5048	37	14	,	,	PUNCT
ejpam-5048	37	15	569	569	NUM
ejpam-5048	37	16	-	-	SYM
ejpam-5048	37	17	581	581	NUM
ejpam-5048	37	18	571	571	NUM
ejpam-5048	37	19	example	example	NOUN
ejpam-5048	37	20	1	1	NUM
ejpam-5048	37	21	.	.	PUNCT
ejpam-5048	38	1	let	let	VERB
ejpam-5048	38	2	x	x	PUNCT
ejpam-5048	38	3	=	=	PRON
ejpam-5048	38	4	{	{	PUNCT
ejpam-5048	38	5	1	1	NUM
ejpam-5048	38	6	,	,	PUNCT
ejpam-5048	38	7	a	a	DET
ejpam-5048	38	8	,	,	PUNCT
ejpam-5048	38	9	b	b	NOUN
ejpam-5048	38	10	,	,	PUNCT
ejpam-5048	38	11	c	c	NOUN
ejpam-5048	38	12	,	,	PUNCT
ejpam-5048	38	13	d	d	NOUN
ejpam-5048	38	14	,	,	PUNCT
ejpam-5048	38	15	e	e	NOUN
ejpam-5048	38	16	,	,	PUNCT
ejpam-5048	38	17	f	f	AUX
ejpam-5048	38	18	}	}	PUNCT
ejpam-5048	38	19	be	be	AUX
ejpam-5048	38	20	a	a	DET
ejpam-5048	38	21	set	set	NOUN
ejpam-5048	38	22	with	with	ADP
ejpam-5048	38	23	the	the	DET
ejpam-5048	38	24	binary	binary	PROPN
ejpam-5048	38	25	operation	operation	NOUN
ejpam-5048	38	26	“	"	PUNCT
ejpam-5048	38	27	∗	∗	NOUN
ejpam-5048	38	28	”	"	PUNCT
ejpam-5048	38	29	in	in	ADP
ejpam-5048	38	30	the	the	DET
ejpam-5048	38	31	following	follow	VERB
ejpam-5048	38	32	cayley	cayley	ADJ
ejpam-5048	38	33	table	table	NOUN
ejpam-5048	38	34	.	.	PUNCT
ejpam-5048	39	1	∗	∗	NOUN
ejpam-5048	39	2	1	1	NUM
ejpam-5048	39	3	a	a	DET
ejpam-5048	39	4	b	b	NOUN
ejpam-5048	39	5	c	c	NOUN
ejpam-5048	39	6	d	d	X
ejpam-5048	39	7	e	e	PROPN
ejpam-5048	39	8	f	f	PROPN
ejpam-5048	39	9	1	1	NUM
ejpam-5048	39	10	1	1	NUM
ejpam-5048	39	11	a	a	DET
ejpam-5048	39	12	b	b	NOUN
ejpam-5048	39	13	c	c	NOUN
ejpam-5048	39	14	d	d	PROPN
ejpam-5048	39	15	e	e	X
ejpam-5048	39	16	f	f	PROPN
ejpam-5048	39	17	a	a	DET
ejpam-5048	39	18	1	1	NUM
ejpam-5048	39	19	1	1	NUM
ejpam-5048	39	20	1	1	NUM
ejpam-5048	39	21	c	c	NOUN
ejpam-5048	39	22	e	e	X
ejpam-5048	39	23	e	e	X
ejpam-5048	39	24	1	1	NUM
ejpam-5048	39	25	b	b	SYM
ejpam-5048	39	26	1	1	NUM
ejpam-5048	39	27	a	a	PRON
ejpam-5048	39	28	1	1	NUM
ejpam-5048	39	29	d	d	NOUN
ejpam-5048	39	30	d	d	PROPN
ejpam-5048	40	1	d	d	X
ejpam-5048	40	2	f	f	PROPN
ejpam-5048	40	3	c	c	NOUN
ejpam-5048	40	4	1	1	NUM
ejpam-5048	40	5	1	1	NUM
ejpam-5048	40	6	b	b	SYM
ejpam-5048	40	7	1	1	NUM
ejpam-5048	40	8	1	1	NUM
ejpam-5048	40	9	1	1	NUM
ejpam-5048	40	10	1	1	NUM
ejpam-5048	40	11	d	d	SYM
ejpam-5048	40	12	1	1	NUM
ejpam-5048	40	13	a	a	DET
ejpam-5048	40	14	1	1	NUM
ejpam-5048	40	15	1	1	NUM
ejpam-5048	40	16	1	1	NUM
ejpam-5048	40	17	1	1	NUM
ejpam-5048	40	18	f	f	NOUN
ejpam-5048	40	19	e	e	ADP
ejpam-5048	40	20	1	1	NUM
ejpam-5048	40	21	a	a	DET
ejpam-5048	40	22	b	b	NUM
ejpam-5048	40	23	1	1	NUM
ejpam-5048	40	24	1	1	NUM
ejpam-5048	40	25	1	1	NUM
ejpam-5048	40	26	1	1	NUM
ejpam-5048	40	27	f	f	NOUN
ejpam-5048	40	28	1	1	NUM
ejpam-5048	40	29	a	a	DET
ejpam-5048	40	30	b	b	X
ejpam-5048	40	31	e	e	X
ejpam-5048	40	32	d	d	PROPN
ejpam-5048	40	33	e	e	PROPN
ejpam-5048	40	34	1	1	NUM
ejpam-5048	40	35	then	then	ADV
ejpam-5048	40	36	x	x	PUNCT
ejpam-5048	40	37	is	be	AUX
ejpam-5048	40	38	a	a	DET
ejpam-5048	40	39	ge	ge	PROPN
ejpam-5048	40	40	-	-	PUNCT
ejpam-5048	40	41	algebra	algebra	PROPN
ejpam-5048	40	42	and	and	CCONJ
ejpam-5048	40	43	we	we	PRON
ejpam-5048	40	44	have	have	VERB
ejpam-5048	40	45	(	(	PUNCT
ejpam-5048	40	46	c	c	NOUN
ejpam-5048	40	47	∗	∗	X
ejpam-5048	40	48	a	a	NOUN
ejpam-5048	40	49	)	)	PUNCT
ejpam-5048	40	50	∗	∗	NOUN
ejpam-5048	40	51	(	(	PUNCT
ejpam-5048	40	52	c	c	NOUN
ejpam-5048	40	53	∗	∗	X
ejpam-5048	40	54	d	d	NOUN
ejpam-5048	40	55	)	)	PUNCT
ejpam-5048	41	1	=	=	SYM
ejpam-5048	41	2	1	1	NUM
ejpam-5048	41	3	∗	∗	NOUN
ejpam-5048	41	4	1	1	NUM
ejpam-5048	41	5	=	=	SYM
ejpam-5048	41	6	1	1	NUM
ejpam-5048	41	7	̸=	̸=	PROPN
ejpam-5048	41	8	e	e	NOUN
ejpam-5048	41	9	=	=	PUNCT
ejpam-5048	41	10	a	a	DET
ejpam-5048	41	11	∗	∗	X
ejpam-5048	41	12	d.	d.	NOUN
ejpam-5048	41	13	we	we	PRON
ejpam-5048	41	14	would	would	AUX
ejpam-5048	41	15	like	like	VERB
ejpam-5048	41	16	to	to	PART
ejpam-5048	41	17	introduce	introduce	VERB
ejpam-5048	41	18	a	a	DET
ejpam-5048	41	19	new	new	ADJ
ejpam-5048	41	20	type	type	NOUN
ejpam-5048	41	21	of	of	ADP
ejpam-5048	41	22	algebra	algebra	NOUN
ejpam-5048	41	23	using	use	VERB
ejpam-5048	41	24	(	(	PUNCT
ejpam-5048	41	25	10	10	NUM
ejpam-5048	41	26	)	)	PUNCT
ejpam-5048	41	27	instead	instead	ADV
ejpam-5048	41	28	of	of	ADP
ejpam-5048	41	29	(	(	PUNCT
ejpam-5048	41	30	ge3	ge3	NOUN
ejpam-5048	41	31	)	)	PUNCT
ejpam-5048	41	32	under	under	ADP
ejpam-5048	41	33	the	the	DET
ejpam-5048	41	34	three	three	NUM
ejpam-5048	41	35	conditions	condition	NOUN
ejpam-5048	41	36	of	of	ADP
ejpam-5048	41	37	ge	ge	PROPN
ejpam-5048	41	38	-	-	PUNCT
ejpam-5048	41	39	agebras	agebras	PROPN
ejpam-5048	41	40	.	.	PUNCT
ejpam-5048	42	1	definition	definition	NOUN
ejpam-5048	42	2	1	1	NUM
ejpam-5048	42	3	.	.	PUNCT
ejpam-5048	43	1	a	a	DET
ejpam-5048	43	2	quasi	quasi	ADJ
ejpam-5048	43	3	ge	ge	PROPN
ejpam-5048	43	4	-	-	PROPN
ejpam-5048	43	5	algebra	algebra	PROPN
ejpam-5048	43	6	(	(	PUNCT
ejpam-5048	43	7	briefly	briefly	ADV
ejpam-5048	43	8	,	,	PUNCT
ejpam-5048	43	9	qge	qge	NOUN
ejpam-5048	43	10	-	-	PUNCT
ejpam-5048	43	11	algebra	algebra	NOUN
ejpam-5048	43	12	)	)	PUNCT
ejpam-5048	43	13	is	be	AUX
ejpam-5048	43	14	defined	define	VERB
ejpam-5048	43	15	to	to	PART
ejpam-5048	43	16	be	be	AUX
ejpam-5048	43	17	a	a	DET
ejpam-5048	43	18	set	set	NOUN
ejpam-5048	43	19	x	x	PUNCT
ejpam-5048	43	20	with	with	ADP
ejpam-5048	43	21	a	a	DET
ejpam-5048	43	22	special	special	ADJ
ejpam-5048	43	23	element	element	NOUN
ejpam-5048	43	24	“	"	PUNCT
ejpam-5048	43	25	1	1	NUM
ejpam-5048	43	26	”	"	PUNCT
ejpam-5048	43	27	called	call	VERB
ejpam-5048	43	28	the	the	DET
ejpam-5048	43	29	unit	unit	NOUN
ejpam-5048	43	30	and	and	CCONJ
ejpam-5048	43	31	a	a	DET
ejpam-5048	43	32	binary	binary	ADJ
ejpam-5048	43	33	operation	operation	NOUN
ejpam-5048	43	34	“	"	PUNCT
ejpam-5048	43	35	∗	∗	NOUN
ejpam-5048	43	36	”	"	PUNCT
ejpam-5048	43	37	that	that	PRON
ejpam-5048	43	38	satisfies	satisfy	VERB
ejpam-5048	43	39	three	three	NUM
ejpam-5048	43	40	conditions	condition	NOUN
ejpam-5048	43	41	(	(	PUNCT
ejpam-5048	43	42	ge1	ge1	NOUN
ejpam-5048	43	43	)	)	PUNCT
ejpam-5048	43	44	,	,	PUNCT
ejpam-5048	43	45	(	(	PUNCT
ejpam-5048	43	46	ge2	ge2	NOUN
ejpam-5048	43	47	)	)	PUNCT
ejpam-5048	43	48	and	and	CCONJ
ejpam-5048	43	49	(	(	PUNCT
ejpam-5048	43	50	10	10	NUM
ejpam-5048	43	51	)	)	PUNCT
ejpam-5048	43	52	.	.	PUNCT
ejpam-5048	44	1	example	example	NOUN
ejpam-5048	45	1	2	2	NUM
ejpam-5048	45	2	.	.	PUNCT
ejpam-5048	45	3	let	let	VERB
ejpam-5048	45	4	x	x	PUNCT
ejpam-5048	45	5	=	=	PRON
ejpam-5048	45	6	{	{	PUNCT
ejpam-5048	45	7	1	1	NUM
ejpam-5048	45	8	,	,	PUNCT
ejpam-5048	45	9	a	a	DET
ejpam-5048	45	10	,	,	PUNCT
ejpam-5048	45	11	b	b	NOUN
ejpam-5048	45	12	,	,	PUNCT
ejpam-5048	45	13	c	c	NOUN
ejpam-5048	45	14	,	,	PUNCT
ejpam-5048	45	15	d	d	NOUN
ejpam-5048	45	16	,	,	PUNCT
ejpam-5048	45	17	e	e	AUX
ejpam-5048	45	18	}	}	PUNCT
ejpam-5048	45	19	be	be	AUX
ejpam-5048	45	20	a	a	DET
ejpam-5048	45	21	set	set	NOUN
ejpam-5048	45	22	with	with	ADP
ejpam-5048	45	23	a	a	DET
ejpam-5048	45	24	binary	binary	ADJ
ejpam-5048	45	25	operation	operation	NOUN
ejpam-5048	45	26	“	"	PUNCT
ejpam-5048	45	27	∗	∗	NOUN
ejpam-5048	45	28	”	"	PUNCT
ejpam-5048	45	29	given	give	VERB
ejpam-5048	45	30	in	in	ADP
ejpam-5048	45	31	the	the	DET
ejpam-5048	45	32	following	follow	VERB
ejpam-5048	45	33	table	table	NOUN
ejpam-5048	45	34	:	:	PUNCT
ejpam-5048	45	35	∗	∗	NOUN
ejpam-5048	45	36	1	1	NUM
ejpam-5048	45	37	a	a	DET
ejpam-5048	45	38	b	b	NOUN
ejpam-5048	45	39	c	c	NOUN
ejpam-5048	45	40	d	d	X
ejpam-5048	45	41	e	e	PROPN
ejpam-5048	45	42	1	1	NUM
ejpam-5048	45	43	1	1	NUM
ejpam-5048	45	44	a	a	DET
ejpam-5048	45	45	b	b	NOUN
ejpam-5048	45	46	c	c	NOUN
ejpam-5048	45	47	d	d	PROPN
ejpam-5048	45	48	e	e	PROPN
ejpam-5048	45	49	a	a	PRON
ejpam-5048	45	50	a	a	DET
ejpam-5048	45	51	1	1	NUM
ejpam-5048	45	52	c	c	NOUN
ejpam-5048	45	53	b	b	PROPN
ejpam-5048	45	54	e	e	PROPN
ejpam-5048	45	55	d	d	PROPN
ejpam-5048	45	56	b	b	PROPN
ejpam-5048	45	57	d	d	X
ejpam-5048	45	58	c	c	PROPN
ejpam-5048	45	59	1	1	NUM
ejpam-5048	45	60	e	e	NOUN
ejpam-5048	45	61	b	b	PROPN
ejpam-5048	45	62	a	a	DET
ejpam-5048	45	63	c	c	NOUN
ejpam-5048	45	64	c	c	NOUN
ejpam-5048	45	65	d	d	X
ejpam-5048	45	66	e	e	PROPN
ejpam-5048	45	67	1	1	NUM
ejpam-5048	45	68	a	a	DET
ejpam-5048	45	69	b	b	PROPN
ejpam-5048	45	70	d	d	X
ejpam-5048	45	71	b	b	PROPN
ejpam-5048	45	72	e	e	X
ejpam-5048	45	73	d	d	X
ejpam-5048	45	74	a	a	DET
ejpam-5048	45	75	1	1	NUM
ejpam-5048	45	76	c	c	NOUN
ejpam-5048	45	77	e	e	X
ejpam-5048	45	78	e	e	X
ejpam-5048	45	79	b	b	PROPN
ejpam-5048	45	80	a	a	PRON
ejpam-5048	45	81	d	d	X
ejpam-5048	45	82	c	c	NOUN
ejpam-5048	45	83	1	1	NUM
ejpam-5048	45	84	it	it	PRON
ejpam-5048	45	85	is	be	AUX
ejpam-5048	45	86	routine	routine	ADJ
ejpam-5048	45	87	to	to	PART
ejpam-5048	45	88	verify	verify	VERB
ejpam-5048	45	89	that	that	SCONJ
ejpam-5048	45	90	(	(	PUNCT
ejpam-5048	45	91	x	x	X
ejpam-5048	45	92	,	,	PUNCT
ejpam-5048	45	93	∗	∗	NOUN
ejpam-5048	45	94	,	,	PUNCT
ejpam-5048	45	95	1	1	NUM
ejpam-5048	45	96	)	)	PUNCT
ejpam-5048	45	97	is	be	AUX
ejpam-5048	45	98	a	a	DET
ejpam-5048	45	99	qge	qge	NOUN
ejpam-5048	45	100	-	-	NOUN
ejpam-5048	45	101	algebra	algebra	NOUN
ejpam-5048	45	102	.	.	PUNCT
ejpam-5048	46	1	example	example	NOUN
ejpam-5048	47	1	3	3	X
ejpam-5048	47	2	.	.	PUNCT
ejpam-5048	47	3	let	let	VERB
ejpam-5048	47	4	x	x	PUNCT
ejpam-5048	47	5	=	=	PRON
ejpam-5048	47	6	{	{	PUNCT
ejpam-5048	47	7	1	1	NUM
ejpam-5048	47	8	,	,	PUNCT
ejpam-5048	47	9	a	a	DET
ejpam-5048	47	10	,	,	PUNCT
ejpam-5048	47	11	b	b	X
ejpam-5048	47	12	}	}	PUNCT
ejpam-5048	47	13	be	be	AUX
ejpam-5048	47	14	a	a	DET
ejpam-5048	47	15	set	set	NOUN
ejpam-5048	47	16	with	with	ADP
ejpam-5048	47	17	the	the	DET
ejpam-5048	47	18	binary	binary	PROPN
ejpam-5048	47	19	operation	operation	NOUN
ejpam-5048	47	20	“	"	PUNCT
ejpam-5048	47	21	∗	∗	NOUN
ejpam-5048	47	22	”	"	PUNCT
ejpam-5048	47	23	in	in	ADP
ejpam-5048	47	24	the	the	DET
ejpam-5048	47	25	following	follow	VERB
ejpam-5048	47	26	cayley	cayley	ADJ
ejpam-5048	47	27	table	table	NOUN
ejpam-5048	47	28	.	.	PUNCT
ejpam-5048	48	1	∗	∗	NOUN
ejpam-5048	48	2	1	1	NUM
ejpam-5048	48	3	a	a	DET
ejpam-5048	48	4	b	b	NUM
ejpam-5048	48	5	1	1	NUM
ejpam-5048	48	6	1	1	NUM
ejpam-5048	48	7	a	a	DET
ejpam-5048	48	8	b	b	NOUN
ejpam-5048	48	9	a	a	PRON
ejpam-5048	48	10	b	b	PROPN
ejpam-5048	48	11	1	1	NUM
ejpam-5048	48	12	a	a	DET
ejpam-5048	48	13	b	b	NOUN
ejpam-5048	48	14	a	a	DET
ejpam-5048	48	15	b	b	NOUN
ejpam-5048	48	16	1	1	NUM
ejpam-5048	48	17	then	then	ADV
ejpam-5048	48	18	x	x	PUNCT
ejpam-5048	48	19	is	be	AUX
ejpam-5048	48	20	a	a	DET
ejpam-5048	48	21	qge	qge	NOUN
ejpam-5048	48	22	-	-	NOUN
ejpam-5048	48	23	algebra	algebra	NOUN
ejpam-5048	48	24	.	.	PUNCT
ejpam-5048	49	1	example	example	NOUN
ejpam-5048	50	1	4	4	NUM
ejpam-5048	50	2	.	.	PUNCT
ejpam-5048	51	1	let	let	VERB
ejpam-5048	51	2	x	x	PRON
ejpam-5048	51	3	be	be	AUX
ejpam-5048	51	4	the	the	DET
ejpam-5048	51	5	set	set	NOUN
ejpam-5048	51	6	of	of	ADP
ejpam-5048	51	7	all	all	DET
ejpam-5048	51	8	integers	integer	NOUN
ejpam-5048	51	9	or	or	CCONJ
ejpam-5048	51	10	all	all	DET
ejpam-5048	51	11	real	real	ADJ
ejpam-5048	51	12	numbers	number	NOUN
ejpam-5048	51	13	.	.	PUNCT
ejpam-5048	52	1	define	define	VERB
ejpam-5048	52	2	a	a	DET
ejpam-5048	52	3	binary	binary	ADJ
ejpam-5048	52	4	operation	operation	NOUN
ejpam-5048	52	5	“	"	PUNCT
ejpam-5048	52	6	∗	∗	NOUN
ejpam-5048	52	7	”	"	PUNCT
ejpam-5048	52	8	on	on	ADP
ejpam-5048	52	9	x	x	PUNCT
ejpam-5048	52	10	as	as	SCONJ
ejpam-5048	52	11	follows	follow	VERB
ejpam-5048	52	12	:	:	PUNCT
ejpam-5048	52	13	∗	∗	NOUN
ejpam-5048	52	14	:	:	PUNCT
ejpam-5048	52	15	x	x	X
ejpam-5048	52	16	×x	×x	X
ejpam-5048	52	17	→	→	SYM
ejpam-5048	52	18	x	x	SYM
ejpam-5048	52	19	,	,	PUNCT
ejpam-5048	52	20	(	(	PUNCT
ejpam-5048	52	21	κ	κ	NOUN
ejpam-5048	52	22	,	,	PUNCT
ejpam-5048	52	23	δ	δ	PROPN
ejpam-5048	52	24	)	)	PUNCT
ejpam-5048	52	25	7→	7→	NUM
ejpam-5048	53	1	δ	δ	NOUN
ejpam-5048	53	2	−	−	NOUN
ejpam-5048	53	3	κ	κ	NOUN
ejpam-5048	53	4	.	.	PUNCT
ejpam-5048	54	1	it	it	PRON
ejpam-5048	54	2	is	be	AUX
ejpam-5048	54	3	routine	routine	ADJ
ejpam-5048	54	4	to	to	PART
ejpam-5048	54	5	verify	verify	VERB
ejpam-5048	54	6	that	that	SCONJ
ejpam-5048	54	7	(	(	PUNCT
ejpam-5048	54	8	x	x	X
ejpam-5048	54	9	,	,	PUNCT
ejpam-5048	54	10	∗	∗	NOUN
ejpam-5048	54	11	,	,	PUNCT
ejpam-5048	54	12	0	0	NUM
ejpam-5048	54	13	)	)	PUNCT
ejpam-5048	54	14	is	be	AUX
ejpam-5048	54	15	a	a	DET
ejpam-5048	54	16	qge	qge	NOUN
ejpam-5048	54	17	-	-	NOUN
ejpam-5048	54	18	algebra	algebra	NOUN
ejpam-5048	54	19	.	.	PUNCT
ejpam-5048	55	1	y.	y.	PROPN
ejpam-5048	55	2	b.	b.	PROPN
ejpam-5048	55	3	jun	jun	PROPN
ejpam-5048	55	4	,	,	PUNCT
ejpam-5048	55	5	ravikumar	ravikumar	PROPN
ejpam-5048	55	6	bandaru	bandaru	PROPN
ejpam-5048	55	7	,	,	PUNCT
ejpam-5048	55	8	rahul	rahul	PROPN
ejpam-5048	55	9	shukla	shukla	PROPN
ejpam-5048	55	10	/	/	SYM
ejpam-5048	55	11	eur	eur	PROPN
ejpam-5048	55	12	.	.	PUNCT
ejpam-5048	56	1	j.	j.	PROPN
ejpam-5048	56	2	pure	pure	PROPN
ejpam-5048	56	3	appl	appl	PROPN
ejpam-5048	56	4	.	.	PROPN
ejpam-5048	56	5	math	math	PROPN
ejpam-5048	56	6	,	,	PUNCT
ejpam-5048	56	7	17	17	NUM
ejpam-5048	56	8	(	(	PUNCT
ejpam-5048	56	9	1	1	NUM
ejpam-5048	56	10	)	)	PUNCT
ejpam-5048	56	11	(	(	PUNCT
ejpam-5048	56	12	2024	2024	NUM
ejpam-5048	56	13	)	)	PUNCT
ejpam-5048	56	14	,	,	PUNCT
ejpam-5048	56	15	569	569	NUM
ejpam-5048	56	16	-	-	SYM
ejpam-5048	56	17	581	581	NUM
ejpam-5048	56	18	572	572	NUM
ejpam-5048	56	19	remark	remark	NOUN
ejpam-5048	56	20	1	1	NUM
ejpam-5048	56	21	.	.	PUNCT
ejpam-5048	56	22	example	example	NOUN
ejpam-5048	56	23	1	1	NUM
ejpam-5048	56	24	explains	explain	VERB
ejpam-5048	56	25	that	that	SCONJ
ejpam-5048	56	26	a	a	DET
ejpam-5048	56	27	ge	ge	PROPN
ejpam-5048	56	28	-	-	PUNCT
ejpam-5048	56	29	algebra	algebra	PROPN
ejpam-5048	56	30	may	may	AUX
ejpam-5048	56	31	not	not	PART
ejpam-5048	56	32	be	be	AUX
ejpam-5048	56	33	a	a	DET
ejpam-5048	56	34	qge	qge	NOUN
ejpam-5048	56	35	-	-	NOUN
ejpam-5048	56	36	algebra	algebra	NOUN
ejpam-5048	56	37	.	.	PUNCT
ejpam-5048	57	1	the	the	DET
ejpam-5048	57	2	following	follow	VERB
ejpam-5048	57	3	example	example	NOUN
ejpam-5048	57	4	shows	show	VERB
ejpam-5048	57	5	that	that	SCONJ
ejpam-5048	57	6	a	a	DET
ejpam-5048	57	7	qge	qge	NOUN
ejpam-5048	57	8	-	-	NOUN
ejpam-5048	57	9	algebra	algebra	NOUN
ejpam-5048	57	10	may	may	AUX
ejpam-5048	57	11	not	not	PART
ejpam-5048	57	12	be	be	AUX
ejpam-5048	57	13	a	a	DET
ejpam-5048	57	14	ge	ge	NOUN
ejpam-5048	57	15	-	-	PUNCT
ejpam-5048	57	16	algebra	algebra	PROPN
ejpam-5048	57	17	.	.	PUNCT
ejpam-5048	58	1	example	example	NOUN
ejpam-5048	58	2	5	5	NUM
ejpam-5048	58	3	.	.	PUNCT
ejpam-5048	59	1	the	the	DET
ejpam-5048	59	2	qge	qge	NOUN
ejpam-5048	59	3	-	-	NOUN
ejpam-5048	59	4	algebra	algebra	NOUN
ejpam-5048	59	5	x	x	PUNCT
ejpam-5048	59	6	given	give	VERB
ejpam-5048	59	7	in	in	ADP
ejpam-5048	59	8	example	example	NOUN
ejpam-5048	59	9	2	2	NUM
ejpam-5048	59	10	is	be	AUX
ejpam-5048	59	11	not	not	PART
ejpam-5048	59	12	a	a	DET
ejpam-5048	59	13	ge	ge	NOUN
ejpam-5048	59	14	-	-	NOUN
ejpam-5048	59	15	algebra	algebra	PROPN
ejpam-5048	59	16	because	because	SCONJ
ejpam-5048	59	17	of	of	ADP
ejpam-5048	59	18	a	a	DET
ejpam-5048	59	19	∗	∗	NOUN
ejpam-5048	59	20	(	(	PUNCT
ejpam-5048	59	21	b	b	NOUN
ejpam-5048	59	22	∗	∗	X
ejpam-5048	59	23	a	a	NOUN
ejpam-5048	59	24	)	)	PUNCT
ejpam-5048	59	25	=	=	PUNCT
ejpam-5048	59	26	a	a	DET
ejpam-5048	59	27	∗	∗	NOUN
ejpam-5048	59	28	c	c	NOUN
ejpam-5048	59	29	=	=	SYM
ejpam-5048	59	30	b	b	X
ejpam-5048	59	31	̸=	̸=	PROPN
ejpam-5048	59	32	e	e	NOUN
ejpam-5048	59	33	=	=	PUNCT
ejpam-5048	59	34	a	a	DET
ejpam-5048	59	35	∗	∗	X
ejpam-5048	59	36	d	d	NOUN
ejpam-5048	59	37	=	=	PUNCT
ejpam-5048	59	38	a	a	DET
ejpam-5048	59	39	∗	∗	NOUN
ejpam-5048	59	40	(	(	PUNCT
ejpam-5048	59	41	b	b	NOUN
ejpam-5048	59	42	∗	∗	NOUN
ejpam-5048	59	43	1	1	NUM
ejpam-5048	59	44	)	)	PUNCT
ejpam-5048	59	45	=	=	PUNCT
ejpam-5048	59	46	a	a	DET
ejpam-5048	59	47	∗	∗	NOUN
ejpam-5048	59	48	(	(	PUNCT
ejpam-5048	59	49	b	b	NOUN
ejpam-5048	59	50	∗	∗	NOUN
ejpam-5048	59	51	(	(	PUNCT
ejpam-5048	59	52	a	a	DET
ejpam-5048	59	53	∗	∗	NOUN
ejpam-5048	59	54	a	a	NOUN
ejpam-5048	59	55	)	)	PUNCT
ejpam-5048	59	56	)	)	PUNCT
ejpam-5048	59	57	.	.	PUNCT
ejpam-5048	60	1	by	by	ADP
ejpam-5048	60	2	remark	remark	NOUN
ejpam-5048	60	3	1	1	NUM
ejpam-5048	60	4	and	and	CCONJ
ejpam-5048	60	5	example	example	NOUN
ejpam-5048	60	6	5	5	NUM
ejpam-5048	60	7	,	,	PUNCT
ejpam-5048	60	8	we	we	PRON
ejpam-5048	60	9	can	can	AUX
ejpam-5048	60	10	see	see	VERB
ejpam-5048	60	11	that	that	SCONJ
ejpam-5048	60	12	the	the	DET
ejpam-5048	60	13	two	two	NUM
ejpam-5048	60	14	concepts	concept	NOUN
ejpam-5048	60	15	ge	ge	PROPN
ejpam-5048	60	16	-	-	PROPN
ejpam-5048	60	17	algebra	algebra	PROPN
ejpam-5048	60	18	and	and	CCONJ
ejpam-5048	60	19	qgealgebra	qgealgebra	NOUN
ejpam-5048	60	20	are	be	AUX
ejpam-5048	60	21	independent	independent	ADJ
ejpam-5048	60	22	of	of	ADP
ejpam-5048	60	23	each	each	DET
ejpam-5048	60	24	other	other	ADJ
ejpam-5048	60	25	.	.	PUNCT
ejpam-5048	61	1	in	in	ADP
ejpam-5048	61	2	a	a	DET
ejpam-5048	61	3	qge	qge	NOUN
ejpam-5048	61	4	-	-	NOUN
ejpam-5048	61	5	algebra	algebra	NOUN
ejpam-5048	61	6	x	x	NOUN
ejpam-5048	61	7	,	,	PUNCT
ejpam-5048	61	8	a	a	DET
ejpam-5048	61	9	binary	binary	ADJ
ejpam-5048	61	10	relation	relation	NOUN
ejpam-5048	61	11	“	"	PUNCT
ejpam-5048	61	12	≤	≤	NUM
ejpam-5048	61	13	”	"	PUNCT
ejpam-5048	61	14	is	be	AUX
ejpam-5048	61	15	also	also	ADV
ejpam-5048	61	16	defined	define	VERB
ejpam-5048	61	17	by	by	ADP
ejpam-5048	61	18	(	(	PUNCT
ejpam-5048	61	19	1	1	NUM
ejpam-5048	61	20	)	)	PUNCT
ejpam-5048	61	21	.	.	PUNCT
ejpam-5048	62	1	if	if	SCONJ
ejpam-5048	62	2	x	x	PRON
ejpam-5048	62	3	is	be	AUX
ejpam-5048	62	4	a	a	DET
ejpam-5048	62	5	ge	ge	PROPN
ejpam-5048	62	6	-	-	PUNCT
ejpam-5048	62	7	algebra	algebra	PROPN
ejpam-5048	62	8	,	,	PUNCT
ejpam-5048	62	9	then	then	ADV
ejpam-5048	62	10	(	(	PUNCT
ejpam-5048	62	11	x,≤	x,≤	X
ejpam-5048	62	12	)	)	PUNCT
ejpam-5048	62	13	may	may	AUX
ejpam-5048	62	14	not	not	PART
ejpam-5048	62	15	be	be	AUX
ejpam-5048	62	16	a	a	DET
ejpam-5048	62	17	poset	poset	NOUN
ejpam-5048	62	18	as	as	SCONJ
ejpam-5048	62	19	shown	show	VERB
ejpam-5048	62	20	in	in	ADP
ejpam-5048	62	21	the	the	DET
ejpam-5048	62	22	following	follow	VERB
ejpam-5048	62	23	example	example	NOUN
ejpam-5048	62	24	.	.	PUNCT
ejpam-5048	63	1	example	example	NOUN
ejpam-5048	64	1	6	6	NUM
ejpam-5048	64	2	.	.	PUNCT
ejpam-5048	65	1	let	let	VERB
ejpam-5048	65	2	x	x	PUNCT
ejpam-5048	65	3	=	=	PRON
ejpam-5048	65	4	{	{	PUNCT
ejpam-5048	65	5	1	1	NUM
ejpam-5048	65	6	,	,	PUNCT
ejpam-5048	65	7	a	a	DET
ejpam-5048	65	8	,	,	PUNCT
ejpam-5048	65	9	b	b	NOUN
ejpam-5048	65	10	,	,	PUNCT
ejpam-5048	65	11	c	c	NOUN
ejpam-5048	65	12	,	,	PUNCT
ejpam-5048	65	13	d	d	AUX
ejpam-5048	65	14	}	}	PUNCT
ejpam-5048	65	15	be	be	AUX
ejpam-5048	65	16	a	a	DET
ejpam-5048	65	17	set	set	NOUN
ejpam-5048	65	18	with	with	ADP
ejpam-5048	65	19	a	a	DET
ejpam-5048	65	20	binary	binary	ADJ
ejpam-5048	65	21	operation	operation	NOUN
ejpam-5048	65	22	“	"	PUNCT
ejpam-5048	65	23	∗	∗	NOUN
ejpam-5048	65	24	”	"	PUNCT
ejpam-5048	65	25	given	give	VERB
ejpam-5048	65	26	in	in	ADP
ejpam-5048	65	27	the	the	DET
ejpam-5048	65	28	following	follow	VERB
ejpam-5048	65	29	table	table	NOUN
ejpam-5048	65	30	:	:	PUNCT
ejpam-5048	65	31	∗	∗	NOUN
ejpam-5048	65	32	1	1	NUM
ejpam-5048	65	33	a	a	DET
ejpam-5048	65	34	b	b	NOUN
ejpam-5048	65	35	c	c	NOUN
ejpam-5048	66	1	d	d	SYM
ejpam-5048	66	2	1	1	NUM
ejpam-5048	66	3	1	1	NUM
ejpam-5048	66	4	a	a	DET
ejpam-5048	66	5	b	b	NOUN
ejpam-5048	66	6	c	c	NOUN
ejpam-5048	66	7	d	d	NOUN
ejpam-5048	66	8	a	a	DET
ejpam-5048	66	9	1	1	NUM
ejpam-5048	66	10	1	1	NUM
ejpam-5048	66	11	1	1	NUM
ejpam-5048	66	12	c	c	NOUN
ejpam-5048	66	13	c	c	NOUN
ejpam-5048	66	14	b	b	SYM
ejpam-5048	66	15	1	1	NUM
ejpam-5048	66	16	a	a	DET
ejpam-5048	66	17	1	1	NUM
ejpam-5048	66	18	d	d	NOUN
ejpam-5048	66	19	d	d	PROPN
ejpam-5048	66	20	c	c	PROPN
ejpam-5048	66	21	1	1	NUM
ejpam-5048	66	22	a	a	DET
ejpam-5048	66	23	1	1	NUM
ejpam-5048	66	24	1	1	NUM
ejpam-5048	66	25	1	1	NUM
ejpam-5048	66	26	d	d	SYM
ejpam-5048	66	27	1	1	NUM
ejpam-5048	66	28	a	a	DET
ejpam-5048	66	29	1	1	NUM
ejpam-5048	66	30	1	1	NUM
ejpam-5048	66	31	1	1	NUM
ejpam-5048	66	32	then	then	ADV
ejpam-5048	66	33	(	(	PUNCT
ejpam-5048	66	34	x	x	X
ejpam-5048	66	35	,	,	PUNCT
ejpam-5048	66	36	∗	∗	NOUN
ejpam-5048	66	37	,	,	PUNCT
ejpam-5048	66	38	1	1	NUM
ejpam-5048	66	39	)	)	PUNCT
ejpam-5048	66	40	is	be	AUX
ejpam-5048	66	41	a	a	DET
ejpam-5048	66	42	ge	ge	NOUN
ejpam-5048	66	43	-	-	NOUN
ejpam-5048	66	44	algebra	algebra	PROPN
ejpam-5048	66	45	.	.	PUNCT
ejpam-5048	67	1	we	we	PRON
ejpam-5048	67	2	can	can	AUX
ejpam-5048	67	3	observe	observe	VERB
ejpam-5048	67	4	that	that	SCONJ
ejpam-5048	67	5	c	c	PROPN
ejpam-5048	67	6	≤	≤	NUM
ejpam-5048	67	7	d	d	NOUN
ejpam-5048	67	8	and	and	CCONJ
ejpam-5048	67	9	d	d	NOUN
ejpam-5048	67	10	≤	≤	NOUN
ejpam-5048	67	11	c	c	NOUN
ejpam-5048	68	1	but	but	CCONJ
ejpam-5048	68	2	c	c	PROPN
ejpam-5048	68	3	̸=	̸=	PROPN
ejpam-5048	68	4	d.	d.	PROPN
ejpam-5048	68	5	hence	hence	ADV
ejpam-5048	68	6	(	(	PUNCT
ejpam-5048	68	7	x,≤	x,≤	X
ejpam-5048	68	8	)	)	PUNCT
ejpam-5048	68	9	is	be	AUX
ejpam-5048	68	10	not	not	PART
ejpam-5048	68	11	be	be	AUX
ejpam-5048	68	12	a	a	DET
ejpam-5048	68	13	poset	poset	NOUN
ejpam-5048	68	14	.	.	PUNCT
ejpam-5048	69	1	but	but	CCONJ
ejpam-5048	69	2	,	,	PUNCT
ejpam-5048	69	3	if	if	SCONJ
ejpam-5048	69	4	x	x	PRON
ejpam-5048	69	5	is	be	AUX
ejpam-5048	69	6	a	a	DET
ejpam-5048	69	7	qge	qge	NOUN
ejpam-5048	69	8	-	-	NOUN
ejpam-5048	69	9	algebra	algebra	NOUN
ejpam-5048	69	10	,	,	PUNCT
ejpam-5048	69	11	then	then	ADV
ejpam-5048	69	12	(	(	PUNCT
ejpam-5048	69	13	x,≤	x,≤	X
ejpam-5048	69	14	)	)	PUNCT
ejpam-5048	69	15	is	be	AUX
ejpam-5048	69	16	a	a	DET
ejpam-5048	69	17	poset	poset	NOUN
ejpam-5048	69	18	.	.	PUNCT
ejpam-5048	70	1	in	in	ADP
ejpam-5048	70	2	fact	fact	NOUN
ejpam-5048	70	3	,	,	PUNCT
ejpam-5048	70	4	it	it	PRON
ejpam-5048	70	5	is	be	AUX
ejpam-5048	70	6	reflexive	reflexive	ADJ
ejpam-5048	70	7	by	by	ADP
ejpam-5048	70	8	(	(	PUNCT
ejpam-5048	70	9	ge1	ge1	NOUN
ejpam-5048	70	10	)	)	PUNCT
ejpam-5048	70	11	.	.	PUNCT
ejpam-5048	71	1	let	let	VERB
ejpam-5048	71	2	κ	κ	VERB
ejpam-5048	71	3	,	,	PUNCT
ejpam-5048	71	4	δ	δ	PROPN
ejpam-5048	71	5	∈	∈	PROPN
ejpam-5048	71	6	x	x	AUX
ejpam-5048	71	7	be	be	AUX
ejpam-5048	71	8	such	such	ADJ
ejpam-5048	71	9	that	that	SCONJ
ejpam-5048	71	10	κ	κ	PROPN
ejpam-5048	71	11	≤	≤	PROPN
ejpam-5048	71	12	δ	δ	PROPN
ejpam-5048	71	13	.	.	PUNCT
ejpam-5048	72	1	then	then	ADV
ejpam-5048	72	2	κ∗	κ∗	PROPN
ejpam-5048	72	3	δ	δ	PROPN
ejpam-5048	72	4	=	=	SYM
ejpam-5048	72	5	1	1	NUM
ejpam-5048	72	6	,	,	PUNCT
ejpam-5048	72	7	and	and	CCONJ
ejpam-5048	72	8	so	so	ADV
ejpam-5048	72	9	δ	δ	PROPN
ejpam-5048	73	1	∗κ	∗κ	PROPN
ejpam-5048	73	2	=	=	PUNCT
ejpam-5048	73	3	(	(	PUNCT
ejpam-5048	73	4	κ∗	κ∗	PROPN
ejpam-5048	73	5	δ)∗	δ)∗	PROPN
ejpam-5048	73	6	(	(	PUNCT
ejpam-5048	73	7	κ∗κ	κ∗κ	NUM
ejpam-5048	73	8	)	)	PUNCT
ejpam-5048	73	9	=	=	SYM
ejpam-5048	73	10	1∗1	1∗1	NUM
ejpam-5048	73	11	=	=	SYM
ejpam-5048	73	12	1	1	NUM
ejpam-5048	73	13	,	,	PUNCT
ejpam-5048	73	14	i.e.	i.e.	X
ejpam-5048	73	15	,	,	PUNCT
ejpam-5048	73	16	δ	δ	PROPN
ejpam-5048	73	17	≤	≤	PROPN
ejpam-5048	73	18	κ	κ	NOUN
ejpam-5048	73	19	.	.	PUNCT
ejpam-5048	74	1	hence	hence	ADV
ejpam-5048	74	2	≤	≤	PROPN
ejpam-5048	74	3	is	be	AUX
ejpam-5048	74	4	symmetric	symmetric	ADJ
ejpam-5048	74	5	.	.	PUNCT
ejpam-5048	75	1	let	let	VERB
ejpam-5048	75	2	κ	κ	NOUN
ejpam-5048	75	3	,	,	PUNCT
ejpam-5048	75	4	δ	δ	PROPN
ejpam-5048	75	5	,	,	PUNCT
ejpam-5048	75	6	ς	ς	PROPN
ejpam-5048	75	7	∈	∈	PROPN
ejpam-5048	75	8	x	x	AUX
ejpam-5048	75	9	be	be	AUX
ejpam-5048	75	10	such	such	ADJ
ejpam-5048	76	1	that	that	SCONJ
ejpam-5048	76	2	κ	κ	PROPN
ejpam-5048	76	3	≤	≤	PROPN
ejpam-5048	76	4	δ	δ	PROPN
ejpam-5048	76	5	and	and	CCONJ
ejpam-5048	76	6	δ	δ	PROPN
ejpam-5048	76	7	≤	≤	PROPN
ejpam-5048	77	1	ς	ς	PROPN
ejpam-5048	77	2	.	.	PUNCT
ejpam-5048	78	1	then	then	ADV
ejpam-5048	78	2	κ	κ	X
ejpam-5048	78	3	∗	∗	NOUN
ejpam-5048	78	4	δ	δ	NOUN
ejpam-5048	79	1	=	=	SYM
ejpam-5048	79	2	1	1	NUM
ejpam-5048	79	3	and	and	CCONJ
ejpam-5048	79	4	δ	δ	PROPN
ejpam-5048	79	5	∗	∗	NOUN
ejpam-5048	79	6	ς	ς	PROPN
ejpam-5048	80	1	=	=	SYM
ejpam-5048	80	2	1	1	X
ejpam-5048	80	3	.	.	PUNCT
ejpam-5048	81	1	hence	hence	ADV
ejpam-5048	81	2	κ	κ	ADP
ejpam-5048	81	3	∗	∗	NOUN
ejpam-5048	81	4	ς	ς	PROPN
ejpam-5048	81	5	=	=	SYM
ejpam-5048	81	6	1	1	NUM
ejpam-5048	81	7	∗	∗	NOUN
ejpam-5048	81	8	(	(	PUNCT
ejpam-5048	81	9	κ	κ	NOUN
ejpam-5048	81	10	∗	∗	NOUN
ejpam-5048	81	11	ς	ς	NOUN
ejpam-5048	81	12	)	)	PUNCT
ejpam-5048	81	13	=	=	SYM
ejpam-5048	81	14	(	(	PUNCT
ejpam-5048	81	15	κ	κ	NOUN
ejpam-5048	81	16	∗	∗	X
ejpam-5048	81	17	δ	δ	PROPN
ejpam-5048	81	18	)	)	PUNCT
ejpam-5048	81	19	∗	∗	NOUN
ejpam-5048	81	20	(	(	PUNCT
ejpam-5048	81	21	κ	κ	NOUN
ejpam-5048	81	22	∗	∗	NOUN
ejpam-5048	81	23	ς	ς	NOUN
ejpam-5048	81	24	)	)	PUNCT
ejpam-5048	82	1	=	=	SYM
ejpam-5048	82	2	δ	δ	PROPN
ejpam-5048	82	3	∗	∗	VERB
ejpam-5048	82	4	ς	ς	PROPN
ejpam-5048	82	5	=	=	SYM
ejpam-5048	82	6	1	1	NUM
ejpam-5048	82	7	,	,	PUNCT
ejpam-5048	82	8	i.e	i.e	PRON
ejpam-5048	82	9	,	,	PUNCT
ejpam-5048	82	10	κ	κ	PROPN
ejpam-5048	82	11	≤	≤	PROPN
ejpam-5048	82	12	ς	ς	NOUN
ejpam-5048	82	13	.	.	PUNCT
ejpam-5048	83	1	thus	thus	ADV
ejpam-5048	83	2	≤	≤	NUM
ejpam-5048	83	3	is	be	AUX
ejpam-5048	83	4	transitive	transitive	ADJ
ejpam-5048	83	5	.	.	PUNCT
ejpam-5048	84	1	therefore	therefore	ADV
ejpam-5048	84	2	(	(	PUNCT
ejpam-5048	84	3	x,≤	x,≤	NOUN
ejpam-5048	84	4	)	)	PUNCT
ejpam-5048	84	5	is	be	AUX
ejpam-5048	84	6	a	a	DET
ejpam-5048	84	7	poset	poset	NOUN
ejpam-5048	84	8	.	.	PUNCT
ejpam-5048	85	1	the	the	DET
ejpam-5048	85	2	relation	relation	NOUN
ejpam-5048	85	3	≤	≤	PROPN
ejpam-5048	85	4	is	be	AUX
ejpam-5048	85	5	also	also	ADV
ejpam-5048	85	6	antisymmetric	antisymmetric	VERB
ejpam-5048	85	7	.	.	PUNCT
ejpam-5048	86	1	in	in	ADP
ejpam-5048	86	2	fact	fact	NOUN
ejpam-5048	86	3	,	,	PUNCT
ejpam-5048	86	4	let	let	VERB
ejpam-5048	86	5	κ	κ	VERB
ejpam-5048	86	6	,	,	PUNCT
ejpam-5048	86	7	δ	δ	PROPN
ejpam-5048	86	8	∈	∈	PROPN
ejpam-5048	86	9	x	x	AUX
ejpam-5048	86	10	be	be	AUX
ejpam-5048	86	11	such	such	ADJ
ejpam-5048	86	12	that	that	SCONJ
ejpam-5048	86	13	κ	κ	PROPN
ejpam-5048	86	14	≤	≤	PROPN
ejpam-5048	86	15	δ	δ	PROPN
ejpam-5048	86	16	and	and	CCONJ
ejpam-5048	86	17	δ	δ	PROPN
ejpam-5048	86	18	≤	≤	PROPN
ejpam-5048	86	19	κ	κ	PROPN
ejpam-5048	86	20	.	.	PUNCT
ejpam-5048	87	1	then	then	ADV
ejpam-5048	87	2	κ	κ	X
ejpam-5048	87	3	∗	∗	NOUN
ejpam-5048	87	4	δ	δ	NOUN
ejpam-5048	88	1	=	=	SYM
ejpam-5048	88	2	1	1	NUM
ejpam-5048	88	3	and	and	CCONJ
ejpam-5048	88	4	δ	δ	PROPN
ejpam-5048	88	5	∗	∗	NOUN
ejpam-5048	88	6	κ	κ	X
ejpam-5048	88	7	=	=	NOUN
ejpam-5048	88	8	1	1	X
ejpam-5048	88	9	.	.	PUNCT
ejpam-5048	88	10	hence	hence	ADV
ejpam-5048	88	11	δ	δ	X
ejpam-5048	88	12	=	=	SYM
ejpam-5048	88	13	1	1	NUM
ejpam-5048	88	14	∗	∗	NOUN
ejpam-5048	88	15	δ	δ	NOUN
ejpam-5048	88	16	=	=	PRON
ejpam-5048	88	17	(	(	PUNCT
ejpam-5048	88	18	δ	δ	PROPN
ejpam-5048	88	19	∗	∗	PROPN
ejpam-5048	88	20	1	1	NUM
ejpam-5048	88	21	)	)	PUNCT
ejpam-5048	88	22	∗	∗	NOUN
ejpam-5048	88	23	(	(	PUNCT
ejpam-5048	88	24	δ	δ	PROPN
ejpam-5048	88	25	∗	∗	X
ejpam-5048	88	26	δ	δ	PROPN
ejpam-5048	88	27	)	)	PUNCT
ejpam-5048	88	28	=	=	PUNCT
ejpam-5048	89	1	(	(	PUNCT
ejpam-5048	89	2	δ	δ	PROPN
ejpam-5048	89	3	∗	∗	PROPN
ejpam-5048	89	4	1	1	NUM
ejpam-5048	89	5	)	)	PUNCT
ejpam-5048	89	6	∗	∗	NOUN
ejpam-5048	89	7	(	(	PUNCT
ejpam-5048	89	8	δ	δ	PROPN
ejpam-5048	89	9	∗	∗	PROPN
ejpam-5048	89	10	κ	κ	NOUN
ejpam-5048	89	11	)	)	PUNCT
ejpam-5048	89	12	=	=	SYM
ejpam-5048	89	13	1	1	NUM
ejpam-5048	89	14	∗	∗	NOUN
ejpam-5048	89	15	κ	κ	X
ejpam-5048	89	16	=	=	SYM
ejpam-5048	89	17	κ	κ	NOUN
ejpam-5048	89	18	,	,	PUNCT
ejpam-5048	89	19	and	and	CCONJ
ejpam-5048	89	20	therefore	therefore	ADV
ejpam-5048	89	21	≤	≤	PROPN
ejpam-5048	89	22	is	be	AUX
ejpam-5048	89	23	antisymmetric	antisymmetric	ADJ
ejpam-5048	89	24	.	.	PUNCT
ejpam-5048	90	1	in	in	ADP
ejpam-5048	90	2	general	general	ADJ
ejpam-5048	90	3	,	,	PUNCT
ejpam-5048	90	4	a	a	DET
ejpam-5048	90	5	ge	ge	PROPN
ejpam-5048	90	6	-	-	PUNCT
ejpam-5048	90	7	algebra	algebra	PROPN
ejpam-5048	90	8	has	have	VERB
ejpam-5048	90	9	no	no	DET
ejpam-5048	90	10	left	left	ADJ
ejpam-5048	90	11	cancellation	cancellation	NOUN
ejpam-5048	90	12	property	property	NOUN
ejpam-5048	90	13	as	as	SCONJ
ejpam-5048	90	14	shown	show	VERB
ejpam-5048	90	15	in	in	ADP
ejpam-5048	90	16	the	the	DET
ejpam-5048	90	17	following	follow	VERB
ejpam-5048	90	18	example	example	NOUN
ejpam-5048	90	19	.	.	PUNCT
ejpam-5048	91	1	example	example	NOUN
ejpam-5048	92	1	7	7	NUM
ejpam-5048	92	2	.	.	PUNCT
ejpam-5048	93	1	the	the	DET
ejpam-5048	93	2	ge	ge	PROPN
ejpam-5048	93	3	-	-	PROPN
ejpam-5048	93	4	algebra	algebra	PROPN
ejpam-5048	93	5	x	x	PUNCT
ejpam-5048	93	6	in	in	ADP
ejpam-5048	93	7	example	example	NOUN
ejpam-5048	93	8	6	6	NUM
ejpam-5048	93	9	does	do	AUX
ejpam-5048	93	10	n’t	not	PART
ejpam-5048	93	11	have	have	VERB
ejpam-5048	93	12	the	the	DET
ejpam-5048	93	13	left	left	ADJ
ejpam-5048	93	14	cancellation	cancellation	NOUN
ejpam-5048	93	15	property	property	NOUN
ejpam-5048	93	16	since	since	SCONJ
ejpam-5048	93	17	a	a	DET
ejpam-5048	93	18	∗	∗	NOUN
ejpam-5048	93	19	a	a	DET
ejpam-5048	93	20	=	=	SYM
ejpam-5048	93	21	1	1	NUM
ejpam-5048	93	22	=	=	SYM
ejpam-5048	93	23	a	a	DET
ejpam-5048	93	24	∗	∗	NOUN
ejpam-5048	93	25	1	1	NUM
ejpam-5048	93	26	,	,	PUNCT
ejpam-5048	93	27	but	but	CCONJ
ejpam-5048	93	28	a	a	DET
ejpam-5048	93	29	̸=	̸=	PROPN
ejpam-5048	93	30	1	1	NUM
ejpam-5048	93	31	.	.	PUNCT
ejpam-5048	94	1	theorem	theorem	NOUN
ejpam-5048	94	2	1	1	NUM
ejpam-5048	94	3	.	.	PUNCT
ejpam-5048	95	1	a	a	DET
ejpam-5048	95	2	qge	qge	NOUN
ejpam-5048	95	3	-	-	NOUN
ejpam-5048	95	4	algebra	algebra	NOUN
ejpam-5048	95	5	x	x	PUNCT
ejpam-5048	95	6	has	have	VERB
ejpam-5048	95	7	the	the	DET
ejpam-5048	95	8	left	left	ADJ
ejpam-5048	95	9	cancellation	cancellation	NOUN
ejpam-5048	95	10	property	property	NOUN
ejpam-5048	95	11	.	.	PUNCT
ejpam-5048	96	1	y.	y.	PROPN
ejpam-5048	96	2	b.	b.	PROPN
ejpam-5048	96	3	jun	jun	PROPN
ejpam-5048	96	4	,	,	PUNCT
ejpam-5048	96	5	ravikumar	ravikumar	PROPN
ejpam-5048	96	6	bandaru	bandaru	PROPN
ejpam-5048	96	7	,	,	PUNCT
ejpam-5048	96	8	rahul	rahul	PROPN
ejpam-5048	96	9	shukla	shukla	PROPN
ejpam-5048	96	10	/	/	SYM
ejpam-5048	96	11	eur	eur	PROPN
ejpam-5048	96	12	.	.	PUNCT
ejpam-5048	97	1	j.	j.	PROPN
ejpam-5048	97	2	pure	pure	PROPN
ejpam-5048	97	3	appl	appl	PROPN
ejpam-5048	97	4	.	.	PROPN
ejpam-5048	97	5	math	math	PROPN
ejpam-5048	97	6	,	,	PUNCT
ejpam-5048	97	7	17	17	NUM
ejpam-5048	97	8	(	(	PUNCT
ejpam-5048	97	9	1	1	NUM
ejpam-5048	97	10	)	)	PUNCT
ejpam-5048	97	11	(	(	PUNCT
ejpam-5048	97	12	2024	2024	NUM
ejpam-5048	97	13	)	)	PUNCT
ejpam-5048	97	14	,	,	PUNCT
ejpam-5048	97	15	569	569	NUM
ejpam-5048	97	16	-	-	SYM
ejpam-5048	97	17	581	581	NUM
ejpam-5048	97	18	573	573	NUM
ejpam-5048	97	19	proof	proof	NOUN
ejpam-5048	97	20	.	.	PUNCT
ejpam-5048	98	1	let	let	VERB
ejpam-5048	98	2	κ	κ	NOUN
ejpam-5048	98	3	,	,	PUNCT
ejpam-5048	98	4	δ	δ	PROPN
ejpam-5048	98	5	,	,	PUNCT
ejpam-5048	98	6	ς	ς	PROPN
ejpam-5048	98	7	∈	∈	PROPN
ejpam-5048	98	8	x	x	AUX
ejpam-5048	98	9	be	be	AUX
ejpam-5048	98	10	such	such	ADJ
ejpam-5048	98	11	that	that	SCONJ
ejpam-5048	98	12	κ	κ	PROPN
ejpam-5048	98	13	∗	∗	X
ejpam-5048	98	14	δ	δ	PROPN
ejpam-5048	99	1	=	=	SYM
ejpam-5048	99	2	κ	κ	PROPN
ejpam-5048	99	3	∗	∗	NOUN
ejpam-5048	99	4	ς	ς	PROPN
ejpam-5048	99	5	.	.	PUNCT
ejpam-5048	100	1	then	then	ADV
ejpam-5048	100	2	δ	δ	X
ejpam-5048	100	3	=	=	SYM
ejpam-5048	100	4	1	1	NUM
ejpam-5048	100	5	∗	∗	NOUN
ejpam-5048	100	6	δ	δ	NOUN
ejpam-5048	100	7	=	=	SYM
ejpam-5048	100	8	(	(	PUNCT
ejpam-5048	100	9	κ	κ	NOUN
ejpam-5048	100	10	∗	∗	NOUN
ejpam-5048	100	11	1	1	NUM
ejpam-5048	100	12	)	)	PUNCT
ejpam-5048	100	13	∗	∗	NOUN
ejpam-5048	100	14	(	(	PUNCT
ejpam-5048	100	15	κ	κ	NOUN
ejpam-5048	100	16	∗	∗	X
ejpam-5048	100	17	δ	δ	PROPN
ejpam-5048	100	18	)	)	PUNCT
ejpam-5048	100	19	=	=	PUNCT
ejpam-5048	101	1	(	(	PUNCT
ejpam-5048	101	2	κ	κ	NOUN
ejpam-5048	101	3	∗	∗	NOUN
ejpam-5048	101	4	1	1	NUM
ejpam-5048	101	5	)	)	PUNCT
ejpam-5048	101	6	∗	∗	NOUN
ejpam-5048	101	7	(	(	PUNCT
ejpam-5048	101	8	κ	κ	NOUN
ejpam-5048	101	9	∗	∗	NOUN
ejpam-5048	101	10	ς	ς	NOUN
ejpam-5048	101	11	)	)	PUNCT
ejpam-5048	101	12	=	=	SYM
ejpam-5048	101	13	1	1	NUM
ejpam-5048	101	14	∗	∗	NOUN
ejpam-5048	101	15	ς	ς	PROPN
ejpam-5048	102	1	=	=	SYM
ejpam-5048	102	2	ς	ς	PROPN
ejpam-5048	102	3	by	by	X
ejpam-5048	102	4	(	(	PUNCT
ejpam-5048	102	5	ge2	ge2	PROPN
ejpam-5048	102	6	)	)	PUNCT
ejpam-5048	102	7	and	and	CCONJ
ejpam-5048	102	8	(	(	PUNCT
ejpam-5048	102	9	10	10	NUM
ejpam-5048	102	10	)	)	PUNCT
ejpam-5048	102	11	.	.	PUNCT
ejpam-5048	103	1	hence	hence	ADV
ejpam-5048	103	2	κ	κ	PROPN
ejpam-5048	103	3	is	be	AUX
ejpam-5048	103	4	left	leave	VERB
ejpam-5048	103	5	-	-	PUNCT
ejpam-5048	103	6	cancellative	cancellative	ADJ
ejpam-5048	103	7	.	.	PUNCT
ejpam-5048	104	1	since	since	SCONJ
ejpam-5048	104	2	κ	κ	PROPN
ejpam-5048	104	3	is	be	AUX
ejpam-5048	104	4	arbitrary	arbitrary	ADJ
ejpam-5048	104	5	,	,	PUNCT
ejpam-5048	104	6	x	x	PUNCT
ejpam-5048	104	7	has	have	VERB
ejpam-5048	104	8	the	the	DET
ejpam-5048	104	9	left	left	ADJ
ejpam-5048	104	10	cancellation	cancellation	NOUN
ejpam-5048	104	11	property	property	NOUN
ejpam-5048	104	12	.	.	PUNCT
ejpam-5048	105	1	proposition	proposition	NOUN
ejpam-5048	105	2	1	1	NUM
ejpam-5048	105	3	.	.	PUNCT
ejpam-5048	106	1	every	every	DET
ejpam-5048	106	2	qge	qge	NOUN
ejpam-5048	106	3	-	-	NOUN
ejpam-5048	106	4	algebra	algebra	NOUN
ejpam-5048	106	5	x	x	SYM
ejpam-5048	106	6	satisfies	satisfie	NOUN
ejpam-5048	106	7	:	:	PUNCT
ejpam-5048	106	8	(	(	PUNCT
ejpam-5048	106	9	∀κ	∀κ	ADV
ejpam-5048	106	10	,	,	PUNCT
ejpam-5048	106	11	δ	δ	PROPN
ejpam-5048	106	12	∈	∈	PROPN
ejpam-5048	106	13	x)(κ	x)(κ	PROPN
ejpam-5048	106	14	≤	≤	NUM
ejpam-5048	106	15	δ	δ	PROPN
ejpam-5048	106	16	⇔	⇔	PROPN
ejpam-5048	106	17	κ	κ	PROPN
ejpam-5048	106	18	=	=	SYM
ejpam-5048	106	19	δ	δ	PROPN
ejpam-5048	106	20	)	)	PUNCT
ejpam-5048	106	21	.	.	PUNCT
ejpam-5048	107	1	(	(	PUNCT
ejpam-5048	107	2	11	11	NUM
ejpam-5048	107	3	)	)	PUNCT
ejpam-5048	107	4	proof	proof	NOUN
ejpam-5048	107	5	.	.	PUNCT
ejpam-5048	108	1	it	it	PRON
ejpam-5048	108	2	is	be	AUX
ejpam-5048	108	3	clear	clear	ADJ
ejpam-5048	108	4	that	that	SCONJ
ejpam-5048	108	5	if	if	SCONJ
ejpam-5048	108	6	κ	κ	PROPN
ejpam-5048	108	7	=	=	SYM
ejpam-5048	108	8	δ	δ	PROPN
ejpam-5048	108	9	,	,	PUNCT
ejpam-5048	108	10	then	then	ADV
ejpam-5048	108	11	κ	κ	X
ejpam-5048	108	12	≤	≤	PROPN
ejpam-5048	108	13	δ	δ	PROPN
ejpam-5048	108	14	.	.	PUNCT
ejpam-5048	109	1	let	let	VERB
ejpam-5048	109	2	κ	κ	NOUN
ejpam-5048	109	3	,	,	PUNCT
ejpam-5048	109	4	δ	δ	PROPN
ejpam-5048	109	5	∈	∈	PROPN
ejpam-5048	109	6	x	x	AUX
ejpam-5048	109	7	be	be	AUX
ejpam-5048	109	8	such	such	ADJ
ejpam-5048	109	9	that	that	SCONJ
ejpam-5048	109	10	κ	κ	PROPN
ejpam-5048	109	11	≤	≤	PROPN
ejpam-5048	109	12	δ	δ	PROPN
ejpam-5048	109	13	.	.	PUNCT
ejpam-5048	110	1	then	then	ADV
ejpam-5048	110	2	κ	κ	X
ejpam-5048	110	3	∗	∗	NOUN
ejpam-5048	110	4	δ	δ	NOUN
ejpam-5048	111	1	=	=	SYM
ejpam-5048	112	1	1	1	NUM
ejpam-5048	112	2	=	=	SYM
ejpam-5048	112	3	κ	κ	X
ejpam-5048	112	4	∗	∗	NOUN
ejpam-5048	112	5	κ	κ	NOUN
ejpam-5048	112	6	by	by	ADP
ejpam-5048	112	7	(	(	PUNCT
ejpam-5048	112	8	ge1	ge1	NOUN
ejpam-5048	112	9	)	)	PUNCT
ejpam-5048	112	10	.	.	PUNCT
ejpam-5048	113	1	it	it	PRON
ejpam-5048	113	2	follows	follow	VERB
ejpam-5048	113	3	from	from	ADP
ejpam-5048	113	4	theorem	theorem	NOUN
ejpam-5048	113	5	1	1	NUM
ejpam-5048	113	6	that	that	DET
ejpam-5048	113	7	κ	κ	AUX
ejpam-5048	113	8	=	=	SYM
ejpam-5048	113	9	δ	δ	PROPN
ejpam-5048	113	10	.	.	PUNCT
ejpam-5048	113	11	remark	remark	PROPN
ejpam-5048	113	12	2	2	NUM
ejpam-5048	113	13	.	.	PUNCT
ejpam-5048	113	14	by	by	ADP
ejpam-5048	113	15	proposition	proposition	NOUN
ejpam-5048	113	16	1	1	NUM
ejpam-5048	113	17	,	,	PUNCT
ejpam-5048	113	18	we	we	PRON
ejpam-5048	113	19	know	know	VERB
ejpam-5048	113	20	that	that	SCONJ
ejpam-5048	113	21	the	the	DET
ejpam-5048	113	22	binary	binary	PROPN
ejpam-5048	113	23	relation	relation	PROPN
ejpam-5048	113	24	≤	≤	NUM
ejpam-5048	113	25	is	be	AUX
ejpam-5048	113	26	only	only	ADV
ejpam-5048	113	27	the	the	DET
ejpam-5048	113	28	set	set	VERB
ejpam-5048	113	29	≤=	≤=	PROPN
ejpam-5048	113	30	{	{	PUNCT
ejpam-5048	113	31	(	(	PUNCT
ejpam-5048	113	32	κ	κ	NOUN
ejpam-5048	113	33	,	,	PUNCT
ejpam-5048	113	34	κ	κ	NOUN
ejpam-5048	113	35	)	)	PUNCT
ejpam-5048	113	36	∈	∈	PROPN
ejpam-5048	113	37	x	x	X
ejpam-5048	113	38	×x	×x	VERB
ejpam-5048	113	39	|	|	ADV
ejpam-5048	113	40	κ	κ	NOUN
ejpam-5048	113	41	∈	∈	NOUN
ejpam-5048	113	42	x	x	X
ejpam-5048	113	43	}	}	PUNCT
ejpam-5048	113	44	.	.	PUNCT
ejpam-5048	114	1	proposition	proposition	NOUN
ejpam-5048	114	2	2	2	NUM
ejpam-5048	114	3	.	.	PUNCT
ejpam-5048	115	1	every	every	DET
ejpam-5048	115	2	qge	qge	NOUN
ejpam-5048	115	3	-	-	NOUN
ejpam-5048	115	4	algebra	algebra	NOUN
ejpam-5048	115	5	x	x	SYM
ejpam-5048	115	6	satisfies	satisfie	NOUN
ejpam-5048	115	7	:	:	PUNCT
ejpam-5048	115	8	(	(	PUNCT
ejpam-5048	115	9	∀κ	∀κ	ADV
ejpam-5048	115	10	,	,	PUNCT
ejpam-5048	115	11	δ	δ	PROPN
ejpam-5048	115	12	∈	∈	PROPN
ejpam-5048	115	13	x)(κ	x)(κ	PROPN
ejpam-5048	115	14	∗	∗	NOUN
ejpam-5048	115	15	δ	δ	PROPN
ejpam-5048	115	16	=	=	PRON
ejpam-5048	115	17	(	(	PUNCT
ejpam-5048	115	18	δ	δ	PROPN
ejpam-5048	115	19	∗	∗	PROPN
ejpam-5048	115	20	κ	κ	NOUN
ejpam-5048	115	21	)	)	PUNCT
ejpam-5048	115	22	∗	∗	NOUN
ejpam-5048	115	23	1	1	NUM
ejpam-5048	115	24	)	)	PUNCT
ejpam-5048	115	25	,	,	PUNCT
ejpam-5048	115	26	(	(	PUNCT
ejpam-5048	115	27	12	12	NUM
ejpam-5048	115	28	)	)	PUNCT
ejpam-5048	115	29	(	(	PUNCT
ejpam-5048	115	30	∀κ	∀κ	ADV
ejpam-5048	115	31	,	,	PUNCT
ejpam-5048	115	32	δ	δ	PROPN
ejpam-5048	115	33	∈	∈	PROPN
ejpam-5048	115	34	x)((κ	x)((κ	PROPN
ejpam-5048	115	35	∗	∗	PROPN
ejpam-5048	115	36	1	1	NUM
ejpam-5048	115	37	)	)	PUNCT
ejpam-5048	115	38	∗	∗	NOUN
ejpam-5048	115	39	(	(	PUNCT
ejpam-5048	115	40	κ	κ	NOUN
ejpam-5048	115	41	∗	∗	X
ejpam-5048	115	42	δ	δ	PROPN
ejpam-5048	115	43	)	)	PUNCT
ejpam-5048	115	44	=	=	SYM
ejpam-5048	115	45	δ	δ	PROPN
ejpam-5048	115	46	)	)	PUNCT
ejpam-5048	115	47	,	,	PUNCT
ejpam-5048	115	48	(	(	PUNCT
ejpam-5048	115	49	13	13	NUM
ejpam-5048	115	50	)	)	PUNCT
ejpam-5048	115	51	(	(	PUNCT
ejpam-5048	115	52	∀κ	∀κ	ADV
ejpam-5048	115	53	,	,	PUNCT
ejpam-5048	115	54	δ	δ	PROPN
ejpam-5048	115	55	∈	∈	PROPN
ejpam-5048	115	56	x)(κ	x)(κ	PROPN
ejpam-5048	115	57	∗	∗	NOUN
ejpam-5048	115	58	(	(	PUNCT
ejpam-5048	115	59	(	(	PUNCT
ejpam-5048	115	60	κ	κ	NOUN
ejpam-5048	115	61	∗	∗	NOUN
ejpam-5048	115	62	1	1	NUM
ejpam-5048	115	63	)	)	PUNCT
ejpam-5048	115	64	∗	∗	PROPN
ejpam-5048	115	65	δ	δ	PROPN
ejpam-5048	115	66	)	)	PUNCT
ejpam-5048	115	67	=	=	SYM
ejpam-5048	115	68	δ	δ	PROPN
ejpam-5048	115	69	)	)	PUNCT
ejpam-5048	115	70	.	.	PUNCT
ejpam-5048	116	1	(	(	PUNCT
ejpam-5048	116	2	14	14	NUM
ejpam-5048	116	3	)	)	PUNCT
ejpam-5048	116	4	(	(	PUNCT
ejpam-5048	116	5	∀κ	∀κ	ADV
ejpam-5048	116	6	,	,	PUNCT
ejpam-5048	116	7	δ	δ	PROPN
ejpam-5048	116	8	,	,	PUNCT
ejpam-5048	116	9	ς	ς	PROPN
ejpam-5048	116	10	∈	∈	PROPN
ejpam-5048	116	11	x)(κ	x)(κ	PUNCT
ejpam-5048	116	12	≤	≤	ADJ
ejpam-5048	116	13	δ	δ	PROPN
ejpam-5048	116	14	⇒	⇒	NOUN
ejpam-5048	116	15	ς	ς	PROPN
ejpam-5048	116	16	∗	∗	NOUN
ejpam-5048	116	17	κ	κ	PROPN
ejpam-5048	116	18	≤	≤	NUM
ejpam-5048	116	19	ς	ς	PROPN
ejpam-5048	116	20	∗	∗	X
ejpam-5048	116	21	δ	δ	PROPN
ejpam-5048	116	22	)	)	PUNCT
ejpam-5048	116	23	.	.	PUNCT
ejpam-5048	117	1	(	(	PUNCT
ejpam-5048	117	2	15	15	X
ejpam-5048	117	3	)	)	PUNCT
ejpam-5048	117	4	proof	proof	NOUN
ejpam-5048	117	5	.	.	PUNCT
ejpam-5048	118	1	the	the	DET
ejpam-5048	118	2	combination	combination	NOUN
ejpam-5048	118	3	of	of	ADP
ejpam-5048	118	4	(	(	PUNCT
ejpam-5048	118	5	ge1	ge1	NOUN
ejpam-5048	118	6	)	)	PUNCT
ejpam-5048	118	7	and	and	CCONJ
ejpam-5048	118	8	(	(	PUNCT
ejpam-5048	118	9	10	10	NUM
ejpam-5048	118	10	)	)	PUNCT
ejpam-5048	118	11	induces	induce	NOUN
ejpam-5048	118	12	(	(	PUNCT
ejpam-5048	118	13	12	12	NUM
ejpam-5048	118	14	)	)	PUNCT
ejpam-5048	118	15	,	,	PUNCT
ejpam-5048	118	16	and	and	CCONJ
ejpam-5048	118	17	the	the	DET
ejpam-5048	118	18	combination	combination	NOUN
ejpam-5048	118	19	of	of	ADP
ejpam-5048	118	20	(	(	PUNCT
ejpam-5048	118	21	ge2	ge2	PROPN
ejpam-5048	118	22	)	)	PUNCT
ejpam-5048	118	23	and	and	CCONJ
ejpam-5048	118	24	(	(	PUNCT
ejpam-5048	118	25	10	10	NUM
ejpam-5048	118	26	)	)	PUNCT
ejpam-5048	118	27	induces	induce	NOUN
ejpam-5048	118	28	(	(	PUNCT
ejpam-5048	118	29	13	13	NUM
ejpam-5048	118	30	)	)	PUNCT
ejpam-5048	118	31	.	.	PUNCT
ejpam-5048	119	1	if	if	SCONJ
ejpam-5048	119	2	we	we	PRON
ejpam-5048	119	3	take	take	VERB
ejpam-5048	119	4	κ	κ	NOUN
ejpam-5048	119	5	=	=	SYM
ejpam-5048	119	6	1	1	NUM
ejpam-5048	119	7	in	in	ADP
ejpam-5048	119	8	(	(	PUNCT
ejpam-5048	119	9	12	12	NUM
ejpam-5048	119	10	)	)	PUNCT
ejpam-5048	119	11	and	and	CCONJ
ejpam-5048	119	12	use	use	NOUN
ejpam-5048	119	13	(	(	PUNCT
ejpam-5048	119	14	ge2	ge2	NOUN
ejpam-5048	119	15	)	)	PUNCT
ejpam-5048	119	16	,	,	PUNCT
ejpam-5048	119	17	then	then	ADV
ejpam-5048	119	18	δ	δ	PROPN
ejpam-5048	119	19	=	=	PRON
ejpam-5048	119	20	(	(	PUNCT
ejpam-5048	119	21	δ	δ	PROPN
ejpam-5048	119	22	∗	∗	PROPN
ejpam-5048	119	23	1	1	NUM
ejpam-5048	119	24	)	)	PUNCT
ejpam-5048	119	25	∗	∗	NOUN
ejpam-5048	119	26	1	1	NUM
ejpam-5048	119	27	for	for	ADP
ejpam-5048	119	28	all	all	DET
ejpam-5048	119	29	δ	δ	PROPN
ejpam-5048	119	30	∈	∈	NOUN
ejpam-5048	119	31	x.	x.	NOUN
ejpam-5048	120	1	it	it	PRON
ejpam-5048	120	2	follows	follow	VERB
ejpam-5048	120	3	from	from	ADP
ejpam-5048	120	4	(	(	PUNCT
ejpam-5048	120	5	13	13	NUM
ejpam-5048	120	6	)	)	PUNCT
ejpam-5048	120	7	that	that	PRON
ejpam-5048	120	8	κ	κ	PROPN
ejpam-5048	120	9	∗	∗	NOUN
ejpam-5048	120	10	(	(	PUNCT
ejpam-5048	120	11	(	(	PUNCT
ejpam-5048	120	12	κ	κ	NOUN
ejpam-5048	120	13	∗	∗	NOUN
ejpam-5048	120	14	1	1	NUM
ejpam-5048	120	15	)	)	PUNCT
ejpam-5048	120	16	∗	∗	PROPN
ejpam-5048	120	17	δ	δ	PROPN
ejpam-5048	120	18	)	)	PUNCT
ejpam-5048	120	19	=	=	SYM
ejpam-5048	121	1	(	(	PUNCT
ejpam-5048	121	2	(	(	PUNCT
ejpam-5048	121	3	κ	κ	NOUN
ejpam-5048	121	4	∗	∗	NOUN
ejpam-5048	121	5	1	1	NUM
ejpam-5048	121	6	)	)	PUNCT
ejpam-5048	121	7	∗	∗	NOUN
ejpam-5048	121	8	1	1	NUM
ejpam-5048	121	9	)	)	PUNCT
ejpam-5048	121	10	∗	∗	NOUN
ejpam-5048	121	11	(	(	PUNCT
ejpam-5048	121	12	(	(	PUNCT
ejpam-5048	121	13	κ	κ	NOUN
ejpam-5048	121	14	∗	∗	NOUN
ejpam-5048	121	15	1	1	NUM
ejpam-5048	121	16	)	)	PUNCT
ejpam-5048	121	17	∗	∗	PROPN
ejpam-5048	121	18	δ	δ	PROPN
ejpam-5048	121	19	)	)	PUNCT
ejpam-5048	121	20	=	=	SYM
ejpam-5048	121	21	δ	δ	PROPN
ejpam-5048	121	22	for	for	ADP
ejpam-5048	121	23	all	all	DET
ejpam-5048	121	24	κ	κ	NOUN
ejpam-5048	121	25	,	,	PUNCT
ejpam-5048	121	26	δ	δ	PROPN
ejpam-5048	121	27	∈	∈	PROPN
ejpam-5048	121	28	x.	x.	NOUN
ejpam-5048	121	29	(	(	PUNCT
ejpam-5048	121	30	15	15	NUM
ejpam-5048	121	31	)	)	PUNCT
ejpam-5048	121	32	is	be	AUX
ejpam-5048	121	33	clear	clear	ADJ
ejpam-5048	121	34	by	by	ADP
ejpam-5048	121	35	(	(	PUNCT
ejpam-5048	121	36	10	10	NUM
ejpam-5048	121	37	)	)	PUNCT
ejpam-5048	121	38	.	.	PUNCT
ejpam-5048	122	1	corollary	corollary	ADJ
ejpam-5048	122	2	1	1	NUM
ejpam-5048	122	3	.	.	PUNCT
ejpam-5048	123	1	every	every	DET
ejpam-5048	123	2	qge	qge	NOUN
ejpam-5048	123	3	-	-	NOUN
ejpam-5048	123	4	algebra	algebra	NOUN
ejpam-5048	123	5	x	x	SYM
ejpam-5048	123	6	satisfies	satisfie	NOUN
ejpam-5048	123	7	:	:	PUNCT
ejpam-5048	123	8	(	(	PUNCT
ejpam-5048	123	9	∀κ	∀κ	ADV
ejpam-5048	123	10	,	,	PUNCT
ejpam-5048	123	11	δ	δ	PROPN
ejpam-5048	123	12	,	,	PUNCT
ejpam-5048	123	13	ς	ς	PROPN
ejpam-5048	123	14	∈	∈	PROPN
ejpam-5048	123	15	x)((ς	x)((ς	NOUN
ejpam-5048	123	16	∗	∗	X
ejpam-5048	123	17	κ	κ	NOUN
ejpam-5048	123	18	)	)	PUNCT
ejpam-5048	123	19	∗	∗	NOUN
ejpam-5048	123	20	(	(	PUNCT
ejpam-5048	123	21	ς	ς	PROPN
ejpam-5048	123	22	∗	∗	X
ejpam-5048	123	23	δ	δ	PROPN
ejpam-5048	123	24	)	)	PUNCT
ejpam-5048	123	25	=	=	PUNCT
ejpam-5048	123	26	(	(	PUNCT
ejpam-5048	123	27	δ	δ	PROPN
ejpam-5048	123	28	∗	∗	PROPN
ejpam-5048	123	29	κ	κ	NOUN
ejpam-5048	123	30	)	)	PUNCT
ejpam-5048	123	31	∗	∗	NOUN
ejpam-5048	123	32	1	1	NUM
ejpam-5048	123	33	)	)	PUNCT
ejpam-5048	123	34	,	,	PUNCT
ejpam-5048	123	35	(	(	PUNCT
ejpam-5048	123	36	16	16	NUM
ejpam-5048	123	37	)	)	PUNCT
ejpam-5048	123	38	(	(	PUNCT
ejpam-5048	123	39	∀κ	∀κ	ADV
ejpam-5048	123	40	,	,	PUNCT
ejpam-5048	123	41	δ	δ	PROPN
ejpam-5048	123	42	,	,	PUNCT
ejpam-5048	123	43	ς	ς	PROPN
ejpam-5048	123	44	∈	∈	PROPN
ejpam-5048	123	45	x)(κ	x)(κ	PROPN
ejpam-5048	123	46	∗	∗	VERB
ejpam-5048	123	47	ς	ς	PROPN
ejpam-5048	123	48	=	=	SYM
ejpam-5048	123	49	δ	δ	PROPN
ejpam-5048	123	50	∗	∗	NOUN
ejpam-5048	123	51	ς	ς	PROPN
ejpam-5048	123	52	⇒	⇒	NOUN
ejpam-5048	123	53	κ	κ	X
ejpam-5048	123	54	=	=	SYM
ejpam-5048	123	55	δ	δ	PROPN
ejpam-5048	123	56	)	)	PUNCT
ejpam-5048	123	57	.	.	PUNCT
ejpam-5048	124	1	(	(	PUNCT
ejpam-5048	124	2	17	17	NUM
ejpam-5048	124	3	)	)	PUNCT
ejpam-5048	124	4	proof	proof	NOUN
ejpam-5048	124	5	.	.	PUNCT
ejpam-5048	125	1	the	the	DET
ejpam-5048	125	2	combination	combination	NOUN
ejpam-5048	125	3	of	of	ADP
ejpam-5048	125	4	(	(	PUNCT
ejpam-5048	125	5	10	10	NUM
ejpam-5048	125	6	)	)	PUNCT
ejpam-5048	125	7	and	and	CCONJ
ejpam-5048	125	8	(	(	PUNCT
ejpam-5048	125	9	12	12	NUM
ejpam-5048	125	10	)	)	PUNCT
ejpam-5048	125	11	induces	induce	NOUN
ejpam-5048	125	12	(	(	PUNCT
ejpam-5048	125	13	16	16	NUM
ejpam-5048	125	14	)	)	PUNCT
ejpam-5048	125	15	.	.	PUNCT
ejpam-5048	126	1	let	let	VERB
ejpam-5048	126	2	κ	κ	NOUN
ejpam-5048	126	3	,	,	PUNCT
ejpam-5048	126	4	δ	δ	PROPN
ejpam-5048	126	5	,	,	PUNCT
ejpam-5048	126	6	ς	ς	PROPN
ejpam-5048	126	7	∈	∈	PROPN
ejpam-5048	126	8	x	x	AUX
ejpam-5048	126	9	be	be	AUX
ejpam-5048	126	10	such	such	ADJ
ejpam-5048	126	11	that	that	SCONJ
ejpam-5048	126	12	κ	κ	PROPN
ejpam-5048	126	13	∗	∗	NOUN
ejpam-5048	126	14	ς	ς	PROPN
ejpam-5048	126	15	=	=	SYM
ejpam-5048	126	16	δ	δ	PROPN
ejpam-5048	126	17	∗	∗	NOUN
ejpam-5048	126	18	ς	ς	PROPN
ejpam-5048	126	19	.	.	PUNCT
ejpam-5048	127	1	using	use	VERB
ejpam-5048	127	2	(	(	PUNCT
ejpam-5048	127	3	12	12	NUM
ejpam-5048	127	4	)	)	PUNCT
ejpam-5048	127	5	,	,	PUNCT
ejpam-5048	127	6	we	we	PRON
ejpam-5048	127	7	have	have	VERB
ejpam-5048	127	8	ς	ς	PROPN
ejpam-5048	127	9	∗	∗	NOUN
ejpam-5048	127	10	κ	κ	NOUN
ejpam-5048	127	11	=	=	SYM
ejpam-5048	127	12	(	(	PUNCT
ejpam-5048	127	13	κ	κ	NOUN
ejpam-5048	127	14	∗	∗	NOUN
ejpam-5048	127	15	ς	ς	NOUN
ejpam-5048	127	16	)	)	PUNCT
ejpam-5048	127	17	∗	∗	NOUN
ejpam-5048	127	18	1	1	NUM
ejpam-5048	127	19	=	=	SYM
ejpam-5048	127	20	(	(	PUNCT
ejpam-5048	127	21	δ	δ	PROPN
ejpam-5048	127	22	∗	∗	PROPN
ejpam-5048	127	23	ς	ς	NOUN
ejpam-5048	127	24	)	)	PUNCT
ejpam-5048	127	25	∗	∗	NOUN
ejpam-5048	127	26	1	1	NUM
ejpam-5048	128	1	=	=	SYM
ejpam-5048	128	2	ς	ς	PROPN
ejpam-5048	128	3	∗	∗	X
ejpam-5048	128	4	δ	δ	PROPN
ejpam-5048	128	5	.	.	PUNCT
ejpam-5048	129	1	it	it	PRON
ejpam-5048	129	2	follows	follow	VERB
ejpam-5048	129	3	from	from	ADP
ejpam-5048	129	4	theorem	theorem	NOUN
ejpam-5048	129	5	1	1	NUM
ejpam-5048	129	6	that	that	PRON
ejpam-5048	129	7	κ	κ	X
ejpam-5048	129	8	=	=	SYM
ejpam-5048	129	9	δ	δ	PROPN
ejpam-5048	129	10	.	.	PUNCT
ejpam-5048	129	11	y.	y.	PROPN
ejpam-5048	129	12	b.	b.	PROPN
ejpam-5048	129	13	jun	jun	PROPN
ejpam-5048	129	14	,	,	PUNCT
ejpam-5048	129	15	ravikumar	ravikumar	PROPN
ejpam-5048	129	16	bandaru	bandaru	PROPN
ejpam-5048	129	17	,	,	PUNCT
ejpam-5048	129	18	rahul	rahul	PROPN
ejpam-5048	129	19	shukla	shukla	PROPN
ejpam-5048	129	20	/	/	SYM
ejpam-5048	129	21	eur	eur	PROPN
ejpam-5048	129	22	.	.	PUNCT
ejpam-5048	130	1	j.	j.	PROPN
ejpam-5048	130	2	pure	pure	PROPN
ejpam-5048	130	3	appl	appl	PROPN
ejpam-5048	130	4	.	.	PROPN
ejpam-5048	130	5	math	math	PROPN
ejpam-5048	130	6	,	,	PUNCT
ejpam-5048	130	7	17	17	NUM
ejpam-5048	130	8	(	(	PUNCT
ejpam-5048	130	9	1	1	NUM
ejpam-5048	130	10	)	)	PUNCT
ejpam-5048	130	11	(	(	PUNCT
ejpam-5048	130	12	2024	2024	NUM
ejpam-5048	130	13	)	)	PUNCT
ejpam-5048	130	14	,	,	PUNCT
ejpam-5048	130	15	569	569	NUM
ejpam-5048	130	16	-	-	SYM
ejpam-5048	130	17	581	581	NUM
ejpam-5048	130	18	574	574	NUM
ejpam-5048	130	19	remark	remark	NOUN
ejpam-5048	130	20	3	3	NUM
ejpam-5048	130	21	.	.	PUNCT
ejpam-5048	131	1	in	in	ADP
ejpam-5048	131	2	proposition	proposition	NOUN
ejpam-5048	131	3	2	2	NUM
ejpam-5048	131	4	,	,	PUNCT
ejpam-5048	131	5	(	(	PUNCT
ejpam-5048	131	6	12	12	NUM
ejpam-5048	131	7	)	)	PUNCT
ejpam-5048	131	8	shows	show	VERB
ejpam-5048	131	9	that	that	SCONJ
ejpam-5048	131	10	x	x	PRON
ejpam-5048	131	11	consists	consist	VERB
ejpam-5048	131	12	of	of	ADP
ejpam-5048	131	13	elements	element	NOUN
ejpam-5048	131	14	κ	κ	ADP
ejpam-5048	131	15	that	that	DET
ejpam-5048	131	16	satisfy	satisfy	NOUN
ejpam-5048	131	17	(	(	PUNCT
ejpam-5048	131	18	κ	κ	NOUN
ejpam-5048	131	19	∗	∗	NOUN
ejpam-5048	131	20	1	1	NUM
ejpam-5048	131	21	)	)	PUNCT
ejpam-5048	131	22	∗	∗	NOUN
ejpam-5048	131	23	1	1	NUM
ejpam-5048	131	24	=	=	SYM
ejpam-5048	131	25	κ	κ	NOUN
ejpam-5048	131	26	.	.	NOUN
ejpam-5048	132	1	that	that	PRON
ejpam-5048	132	2	is	be	AUX
ejpam-5048	132	3	,	,	PUNCT
ejpam-5048	132	4	x	x	X
ejpam-5048	132	5	=	=	PRON
ejpam-5048	132	6	{	{	PUNCT
ejpam-5048	132	7	κ	κ	NOUN
ejpam-5048	132	8	∈	∈	PROPN
ejpam-5048	133	1	x	x	INTJ
ejpam-5048	133	2	|	|	ADV
ejpam-5048	133	3	(	(	PUNCT
ejpam-5048	133	4	κ	κ	NOUN
ejpam-5048	133	5	∗	∗	NOUN
ejpam-5048	133	6	1	1	NUM
ejpam-5048	133	7	)	)	PUNCT
ejpam-5048	133	8	∗	∗	NOUN
ejpam-5048	133	9	1	1	NUM
ejpam-5048	133	10	=	=	SYM
ejpam-5048	133	11	κ	κ	NOUN
ejpam-5048	133	12	}	}	PUNCT
ejpam-5048	133	13	.	.	PUNCT
ejpam-5048	134	1	theorem	theorem	NOUN
ejpam-5048	134	2	2	2	NUM
ejpam-5048	134	3	.	.	X
ejpam-5048	135	1	let	let	VERB
ejpam-5048	135	2	(	(	PUNCT
ejpam-5048	135	3	x	x	NOUN
ejpam-5048	135	4	,	,	PUNCT
ejpam-5048	135	5	∗x	∗x	NOUN
ejpam-5048	135	6	,	,	PUNCT
ejpam-5048	135	7	1x	1x	NUM
ejpam-5048	135	8	)	)	PUNCT
ejpam-5048	135	9	and	and	CCONJ
ejpam-5048	135	10	(	(	PUNCT
ejpam-5048	135	11	y	y	PROPN
ejpam-5048	135	12	,	,	PUNCT
ejpam-5048	135	13	∗y	∗y	PROPN
ejpam-5048	135	14	,	,	PUNCT
ejpam-5048	135	15	1y	1y	PROPN
ejpam-5048	135	16	)	)	PUNCT
ejpam-5048	135	17	be	be	AUX
ejpam-5048	135	18	qge	qge	NOUN
ejpam-5048	135	19	-	-	PUNCT
ejpam-5048	135	20	algebras	algebras	X
ejpam-5048	135	21	.	.	PUNCT
ejpam-5048	136	1	let	let	VERB
ejpam-5048	136	2	z	z	NOUN
ejpam-5048	137	1	=	=	PUNCT
ejpam-5048	138	1	x	x	SYM
ejpam-5048	138	2	×	×	NOUN
ejpam-5048	138	3	y	y	NOUN
ejpam-5048	138	4	be	be	AUX
ejpam-5048	138	5	the	the	DET
ejpam-5048	138	6	cartesian	cartesian	ADJ
ejpam-5048	138	7	product	product	NOUN
ejpam-5048	138	8	of	of	ADP
ejpam-5048	138	9	x	x	PROPN
ejpam-5048	138	10	and	and	CCONJ
ejpam-5048	138	11	y	y	PROPN
ejpam-5048	138	12	.	.	PUNCT
ejpam-5048	139	1	define	define	VERB
ejpam-5048	139	2	a	a	DET
ejpam-5048	139	3	binary	binary	ADJ
ejpam-5048	139	4	operation	operation	NOUN
ejpam-5048	139	5	“	"	PUNCT
ejpam-5048	139	6	∗	∗	NOUN
ejpam-5048	139	7	”	"	PUNCT
ejpam-5048	139	8	on	on	ADP
ejpam-5048	139	9	z	z	PROPN
ejpam-5048	139	10	as	as	SCONJ
ejpam-5048	139	11	follows	follow	VERB
ejpam-5048	139	12	:	:	PUNCT
ejpam-5048	139	13	∗	∗	NOUN
ejpam-5048	139	14	:	:	PUNCT
ejpam-5048	139	15	z	z	NOUN
ejpam-5048	139	16	×	×	NOUN
ejpam-5048	139	17	z	z	PROPN
ejpam-5048	139	18	→	→	SYM
ejpam-5048	139	19	z	z	NOUN
ejpam-5048	139	20	,	,	PUNCT
ejpam-5048	139	21	(	(	PUNCT
ejpam-5048	139	22	(	(	PUNCT
ejpam-5048	139	23	κ,ϖ	κ,ϖ	NOUN
ejpam-5048	139	24	)	)	PUNCT
ejpam-5048	139	25	,	,	PUNCT
ejpam-5048	139	26	(	(	PUNCT
ejpam-5048	139	27	δ	δ	PROPN
ejpam-5048	139	28	,	,	PUNCT
ejpam-5048	139	29	π	π	PROPN
ejpam-5048	139	30	)	)	PUNCT
ejpam-5048	139	31	)	)	PUNCT
ejpam-5048	140	1	7→	7→	NUM
ejpam-5048	140	2	(	(	PUNCT
ejpam-5048	140	3	κ	κ	X
ejpam-5048	140	4	∗x	∗x	PROPN
ejpam-5048	140	5	δ,ϖ	δ,ϖ	PROPN
ejpam-5048	140	6	∗y	∗y	PROPN
ejpam-5048	140	7	π	π	PROPN
ejpam-5048	140	8	)	)	PUNCT
ejpam-5048	140	9	.	.	PUNCT
ejpam-5048	141	1	(	(	PUNCT
ejpam-5048	141	2	18	18	NUM
ejpam-5048	141	3	)	)	PUNCT
ejpam-5048	141	4	then	then	ADV
ejpam-5048	141	5	(	(	PUNCT
ejpam-5048	141	6	z	z	NOUN
ejpam-5048	141	7	,	,	PUNCT
ejpam-5048	141	8	∗	∗	NOUN
ejpam-5048	141	9	,	,	PUNCT
ejpam-5048	141	10	1	1	NUM
ejpam-5048	141	11	)	)	PUNCT
ejpam-5048	141	12	is	be	AUX
ejpam-5048	141	13	a	a	DET
ejpam-5048	141	14	qge	qge	NOUN
ejpam-5048	141	15	-	-	NOUN
ejpam-5048	141	16	algebra	algebra	NOUN
ejpam-5048	141	17	where	where	SCONJ
ejpam-5048	141	18	1	1	NUM
ejpam-5048	141	19	=	=	SYM
ejpam-5048	141	20	(	(	PUNCT
ejpam-5048	141	21	1x	1x	NUM
ejpam-5048	141	22	,	,	PUNCT
ejpam-5048	141	23	1y	1y	PROPN
ejpam-5048	141	24	)	)	PUNCT
ejpam-5048	141	25	.	.	PUNCT
ejpam-5048	142	1	we	we	PRON
ejpam-5048	142	2	call	call	VERB
ejpam-5048	142	3	it	it	PRON
ejpam-5048	142	4	the	the	DET
ejpam-5048	142	5	product	product	NOUN
ejpam-5048	142	6	qge	qge	NOUN
ejpam-5048	142	7	-	-	NOUN
ejpam-5048	142	8	algebra	algebra	NOUN
ejpam-5048	142	9	of	of	ADP
ejpam-5048	142	10	(	(	PUNCT
ejpam-5048	142	11	x	x	NOUN
ejpam-5048	142	12	,	,	PUNCT
ejpam-5048	142	13	∗x	∗x	NOUN
ejpam-5048	142	14	,	,	PUNCT
ejpam-5048	142	15	1x	1x	NUM
ejpam-5048	142	16	)	)	PUNCT
ejpam-5048	142	17	and	and	CCONJ
ejpam-5048	142	18	(	(	PUNCT
ejpam-5048	142	19	y	y	PROPN
ejpam-5048	142	20	,	,	PUNCT
ejpam-5048	142	21	∗y	∗y	PROPN
ejpam-5048	142	22	,	,	PUNCT
ejpam-5048	142	23	1y	1y	PROPN
ejpam-5048	142	24	)	)	PUNCT
ejpam-5048	142	25	.	.	PUNCT
ejpam-5048	143	1	proof	proof	NOUN
ejpam-5048	143	2	.	.	PUNCT
ejpam-5048	144	1	it	it	PRON
ejpam-5048	144	2	is	be	AUX
ejpam-5048	144	3	straightforward	straightforward	ADJ
ejpam-5048	144	4	.	.	PUNCT
ejpam-5048	145	1	an	an	DET
ejpam-5048	145	2	example	example	NOUN
ejpam-5048	145	3	to	to	PART
ejpam-5048	145	4	explain	explain	VERB
ejpam-5048	145	5	theorem	theorem	ADJ
ejpam-5048	145	6	2	2	NUM
ejpam-5048	145	7	is	be	AUX
ejpam-5048	145	8	presented	present	VERB
ejpam-5048	145	9	as	as	SCONJ
ejpam-5048	145	10	follows	follow	VERB
ejpam-5048	145	11	.	.	PUNCT
ejpam-5048	146	1	example	example	NOUN
ejpam-5048	146	2	8	8	NUM
ejpam-5048	146	3	.	.	PUNCT
ejpam-5048	147	1	let	let	VERB
ejpam-5048	147	2	(	(	PUNCT
ejpam-5048	147	3	x	x	NOUN
ejpam-5048	147	4	,	,	PUNCT
ejpam-5048	147	5	∗x	∗x	PROPN
ejpam-5048	147	6	,	,	PUNCT
ejpam-5048	147	7	1x	1x	NUM
ejpam-5048	147	8	)	)	PUNCT
ejpam-5048	147	9	be	be	AUX
ejpam-5048	147	10	a	a	DET
ejpam-5048	147	11	qge	qge	NOUN
ejpam-5048	147	12	-	-	NOUN
ejpam-5048	147	13	algebra	algebra	NOUN
ejpam-5048	147	14	and	and	CCONJ
ejpam-5048	147	15	consider	consider	VERB
ejpam-5048	147	16	the	the	DET
ejpam-5048	147	17	qge	qge	NOUN
ejpam-5048	147	18	-	-	NOUN
ejpam-5048	147	19	algebra	algebra	PROPN
ejpam-5048	147	20	(	(	PUNCT
ejpam-5048	147	21	z,−	z,−	PROPN
ejpam-5048	147	22	,	,	PUNCT
ejpam-5048	147	23	0	0	NUM
ejpam-5048	147	24	)	)	PUNCT
ejpam-5048	147	25	which	which	PRON
ejpam-5048	147	26	is	be	AUX
ejpam-5048	147	27	given	give	VERB
ejpam-5048	147	28	in	in	ADP
ejpam-5048	147	29	example	example	NOUN
ejpam-5048	147	30	4	4	NUM
ejpam-5048	147	31	.	.	PUNCT
ejpam-5048	148	1	let	let	VERB
ejpam-5048	148	2	y	y	NOUN
ejpam-5048	148	3	=	=	PUNCT
ejpam-5048	148	4	x	x	SYM
ejpam-5048	148	5	×	×	PROPN
ejpam-5048	148	6	z	z	NOUN
ejpam-5048	148	7	and	and	CCONJ
ejpam-5048	148	8	the	the	DET
ejpam-5048	148	9	binary	binary	PROPN
ejpam-5048	148	10	operation	operation	NOUN
ejpam-5048	148	11	“	"	PUNCT
ejpam-5048	148	12	∗	∗	NOUN
ejpam-5048	148	13	”	"	PUNCT
ejpam-5048	148	14	on	on	ADP
ejpam-5048	148	15	y	y	PROPN
ejpam-5048	148	16	is	be	AUX
ejpam-5048	148	17	given	give	VERB
ejpam-5048	148	18	as	as	SCONJ
ejpam-5048	148	19	follows	follow	VERB
ejpam-5048	148	20	:	:	PUNCT
ejpam-5048	148	21	(	(	PUNCT
ejpam-5048	148	22	κ,ϖ	κ,ϖ	NOUN
ejpam-5048	148	23	)	)	PUNCT
ejpam-5048	148	24	∗	∗	NOUN
ejpam-5048	148	25	(	(	PUNCT
ejpam-5048	148	26	δ	δ	PROPN
ejpam-5048	148	27	,	,	PUNCT
ejpam-5048	148	28	π	π	PROPN
ejpam-5048	148	29	)	)	PUNCT
ejpam-5048	149	1	=	=	SYM
ejpam-5048	149	2	(	(	PUNCT
ejpam-5048	149	3	κ	κ	X
ejpam-5048	149	4	∗x	∗x	PROPN
ejpam-5048	149	5	δ	δ	PROPN
ejpam-5048	149	6	,	,	PUNCT
ejpam-5048	149	7	π	π	NOUN
ejpam-5048	149	8	−ϖ	−ϖ	NOUN
ejpam-5048	149	9	)	)	PUNCT
ejpam-5048	149	10	for	for	ADP
ejpam-5048	149	11	all	all	DET
ejpam-5048	149	12	(	(	PUNCT
ejpam-5048	149	13	κ,ϖ	κ,ϖ	NOUN
ejpam-5048	149	14	)	)	PUNCT
ejpam-5048	149	15	,	,	PUNCT
ejpam-5048	149	16	(	(	PUNCT
ejpam-5048	149	17	δ	δ	PROPN
ejpam-5048	149	18	,	,	PUNCT
ejpam-5048	149	19	π	π	PROPN
ejpam-5048	149	20	)	)	PUNCT
ejpam-5048	149	21	∈	∈	PROPN
ejpam-5048	149	22	y	y	PROPN
ejpam-5048	149	23	.	.	PUNCT
ejpam-5048	150	1	then	then	ADV
ejpam-5048	150	2	(	(	PUNCT
ejpam-5048	150	3	y	y	NOUN
ejpam-5048	150	4	,	,	PUNCT
ejpam-5048	150	5	∗	∗	NOUN
ejpam-5048	150	6	,	,	PUNCT
ejpam-5048	150	7	1	1	NUM
ejpam-5048	150	8	)	)	PUNCT
ejpam-5048	150	9	is	be	AUX
ejpam-5048	150	10	the	the	DET
ejpam-5048	150	11	product	product	NOUN
ejpam-5048	150	12	qge	qge	NOUN
ejpam-5048	150	13	-	-	NOUN
ejpam-5048	150	14	algebra	algebra	NOUN
ejpam-5048	150	15	of	of	ADP
ejpam-5048	150	16	(	(	PUNCT
ejpam-5048	150	17	x	x	NOUN
ejpam-5048	150	18	,	,	PUNCT
ejpam-5048	150	19	∗x	∗x	NOUN
ejpam-5048	150	20	,	,	PUNCT
ejpam-5048	150	21	1x	1x	NUM
ejpam-5048	150	22	)	)	PUNCT
ejpam-5048	150	23	and	and	CCONJ
ejpam-5048	150	24	(	(	PUNCT
ejpam-5048	150	25	z,−	z,−	PROPN
ejpam-5048	150	26	,	,	PUNCT
ejpam-5048	150	27	0	0	NUM
ejpam-5048	150	28	)	)	PUNCT
ejpam-5048	150	29	where	where	SCONJ
ejpam-5048	150	30	1	1	X
ejpam-5048	150	31	=	=	SYM
ejpam-5048	150	32	(	(	PUNCT
ejpam-5048	150	33	1x	1x	NUM
ejpam-5048	150	34	,	,	PUNCT
ejpam-5048	150	35	0	0	NUM
ejpam-5048	150	36	)	)	PUNCT
ejpam-5048	150	37	.	.	PUNCT
ejpam-5048	151	1	4	4	X
ejpam-5048	151	2	.	.	X
ejpam-5048	151	3	qge	qge	NOUN
ejpam-5048	151	4	-	-	PUNCT
ejpam-5048	151	5	subalgebras	subalgebras	PROPN
ejpam-5048	151	6	in	in	ADP
ejpam-5048	151	7	what	what	PRON
ejpam-5048	151	8	follows	follow	VERB
ejpam-5048	151	9	,	,	PUNCT
ejpam-5048	151	10	let	let	VERB
ejpam-5048	151	11	x	x	PRON
ejpam-5048	151	12	be	be	AUX
ejpam-5048	151	13	a	a	DET
ejpam-5048	151	14	qge	qge	NOUN
ejpam-5048	151	15	-	-	NOUN
ejpam-5048	151	16	algebra	algebra	NOUN
ejpam-5048	151	17	unless	unless	SCONJ
ejpam-5048	151	18	otherwise	otherwise	ADV
ejpam-5048	151	19	specified	specify	VERB
ejpam-5048	151	20	.	.	PUNCT
ejpam-5048	152	1	definition	definition	NOUN
ejpam-5048	152	2	2	2	NUM
ejpam-5048	152	3	.	.	PUNCT
ejpam-5048	153	1	a	a	DET
ejpam-5048	153	2	non	non	ADJ
ejpam-5048	153	3	-	-	ADJ
ejpam-5048	153	4	empty	empty	ADJ
ejpam-5048	153	5	subset	subset	NOUN
ejpam-5048	153	6	e	e	NOUN
ejpam-5048	153	7	of	of	ADP
ejpam-5048	153	8	x	x	PROPN
ejpam-5048	153	9	is	be	AUX
ejpam-5048	153	10	called	call	VERB
ejpam-5048	153	11	a	a	DET
ejpam-5048	153	12	qge	qge	NOUN
ejpam-5048	153	13	-	-	PUNCT
ejpam-5048	153	14	subalgebra	subalgebra	NOUN
ejpam-5048	153	15	of	of	ADP
ejpam-5048	153	16	x	x	PRON
ejpam-5048	153	17	if	if	SCONJ
ejpam-5048	153	18	it	it	PRON
ejpam-5048	153	19	satisfies	satisfy	VERB
ejpam-5048	153	20	:	:	PUNCT
ejpam-5048	153	21	(	(	PUNCT
ejpam-5048	153	22	∀κ	∀κ	ADV
ejpam-5048	153	23	,	,	PUNCT
ejpam-5048	153	24	δ	δ	PROPN
ejpam-5048	153	25	∈	∈	PROPN
ejpam-5048	153	26	x)(κ	x)(κ	PROPN
ejpam-5048	153	27	,	,	PUNCT
ejpam-5048	153	28	δ	δ	PROPN
ejpam-5048	153	29	∈	∈	PROPN
ejpam-5048	153	30	e	e	PROPN
ejpam-5048	153	31	⇒	⇒	NOUN
ejpam-5048	153	32	κ	κ	PROPN
ejpam-5048	153	33	∗	∗	NOUN
ejpam-5048	153	34	δ	δ	PROPN
ejpam-5048	153	35	∈	∈	PROPN
ejpam-5048	153	36	e	e	PROPN
ejpam-5048	153	37	)	)	PUNCT
ejpam-5048	153	38	.	.	PUNCT
ejpam-5048	154	1	(	(	PUNCT
ejpam-5048	154	2	19	19	NUM
ejpam-5048	154	3	)	)	PUNCT
ejpam-5048	154	4	it	it	PRON
ejpam-5048	154	5	is	be	AUX
ejpam-5048	154	6	obvious	obvious	ADJ
ejpam-5048	154	7	that	that	SCONJ
ejpam-5048	154	8	the	the	DET
ejpam-5048	154	9	singleton	singleton	NOUN
ejpam-5048	154	10	{	{	PUNCT
ejpam-5048	154	11	1	1	NUM
ejpam-5048	154	12	}	}	PUNCT
ejpam-5048	154	13	is	be	AUX
ejpam-5048	154	14	a	a	DET
ejpam-5048	154	15	qge	qge	NOUN
ejpam-5048	154	16	-	-	PUNCT
ejpam-5048	154	17	subalgebra	subalgebra	NOUN
ejpam-5048	154	18	of	of	ADP
ejpam-5048	154	19	x.	x.	NOUN
ejpam-5048	154	20	example	example	NOUN
ejpam-5048	154	21	9	9	NUM
ejpam-5048	154	22	.	.	PUNCT
ejpam-5048	155	1	consider	consider	VERB
ejpam-5048	155	2	the	the	DET
ejpam-5048	155	3	qge	qge	NOUN
ejpam-5048	155	4	-	-	NOUN
ejpam-5048	155	5	algebra	algebra	NOUN
ejpam-5048	155	6	x	x	PUNCT
ejpam-5048	155	7	given	give	VERB
ejpam-5048	155	8	in	in	ADP
ejpam-5048	155	9	example	example	NOUN
ejpam-5048	155	10	2	2	X
ejpam-5048	155	11	.	.	PUNCT
ejpam-5048	156	1	it	it	PRON
ejpam-5048	156	2	is	be	AUX
ejpam-5048	156	3	routine	routine	ADJ
ejpam-5048	156	4	to	to	PART
ejpam-5048	156	5	verify	verify	VERB
ejpam-5048	156	6	that	that	SCONJ
ejpam-5048	156	7	the	the	DET
ejpam-5048	156	8	set	set	NOUN
ejpam-5048	156	9	e	e	NOUN
ejpam-5048	156	10	=	=	PUNCT
ejpam-5048	156	11	{	{	PUNCT
ejpam-5048	156	12	1	1	NUM
ejpam-5048	156	13	,	,	PUNCT
ejpam-5048	156	14	b	b	NOUN
ejpam-5048	156	15	,	,	PUNCT
ejpam-5048	156	16	d	d	NOUN
ejpam-5048	156	17	}	}	PUNCT
ejpam-5048	156	18	is	be	AUX
ejpam-5048	156	19	a	a	DET
ejpam-5048	156	20	qge	qge	NOUN
ejpam-5048	156	21	-	-	PUNCT
ejpam-5048	156	22	subalgebra	subalgebra	NOUN
ejpam-5048	156	23	of	of	ADP
ejpam-5048	156	24	x.	x.	PROPN
ejpam-5048	156	25	example	example	NOUN
ejpam-5048	157	1	10	10	NUM
ejpam-5048	157	2	.	.	PUNCT
ejpam-5048	158	1	let	let	VERB
ejpam-5048	158	2	x	x	PRON
ejpam-5048	158	3	:	:	PUNCT
ejpam-5048	158	4	=	=	SYM
ejpam-5048	158	5	r	r	NOUN
ejpam-5048	158	6	\	\	PUNCT
ejpam-5048	158	7	{	{	PUNCT
ejpam-5048	158	8	0	0	NUM
ejpam-5048	158	9	}	}	PUNCT
ejpam-5048	158	10	where	where	SCONJ
ejpam-5048	158	11	r	r	NOUN
ejpam-5048	158	12	is	be	AUX
ejpam-5048	158	13	the	the	DET
ejpam-5048	158	14	set	set	NOUN
ejpam-5048	158	15	of	of	ADP
ejpam-5048	158	16	all	all	DET
ejpam-5048	158	17	real	real	ADJ
ejpam-5048	158	18	numbers	number	NOUN
ejpam-5048	158	19	.	.	PUNCT
ejpam-5048	159	1	define	define	VERB
ejpam-5048	159	2	binary	binary	ADJ
ejpam-5048	159	3	operations	operation	NOUN
ejpam-5048	159	4	“	"	PUNCT
ejpam-5048	159	5	∗+	∗+	PROPN
ejpam-5048	159	6	”	"	PUNCT
ejpam-5048	159	7	and	and	CCONJ
ejpam-5048	159	8	“	"	PUNCT
ejpam-5048	159	9	∗−	∗−	ADJ
ejpam-5048	159	10	”	"	PUNCT
ejpam-5048	159	11	on	on	ADP
ejpam-5048	159	12	x	x	PUNCT
ejpam-5048	159	13	as	as	SCONJ
ejpam-5048	159	14	follows	follow	VERB
ejpam-5048	159	15	:	:	PUNCT
ejpam-5048	159	16	∗+	∗+	NOUN
ejpam-5048	159	17	:	:	PUNCT
ejpam-5048	159	18	x	x	X
ejpam-5048	159	19	×x	×x	X
ejpam-5048	159	20	→	→	SYM
ejpam-5048	159	21	x	x	SYM
ejpam-5048	159	22	,	,	PUNCT
ejpam-5048	159	23	(	(	PUNCT
ejpam-5048	159	24	κ	κ	NOUN
ejpam-5048	159	25	,	,	PUNCT
ejpam-5048	159	26	δ	δ	PROPN
ejpam-5048	159	27	)	)	PUNCT
ejpam-5048	159	28	7→	7→	NUM
ejpam-5048	160	1	δ	δ	PROPN
ejpam-5048	160	2	κ	κ	NOUN
ejpam-5048	160	3	,	,	PUNCT
ejpam-5048	160	4	(	(	PUNCT
ejpam-5048	160	5	20	20	NUM
ejpam-5048	160	6	)	)	PUNCT
ejpam-5048	160	7	∗−	∗−	NOUN
ejpam-5048	160	8	:	:	PUNCT
ejpam-5048	160	9	x	x	X
ejpam-5048	160	10	×x	×x	X
ejpam-5048	160	11	→	→	SYM
ejpam-5048	160	12	x	x	SYM
ejpam-5048	160	13	,	,	PUNCT
ejpam-5048	160	14	(	(	PUNCT
ejpam-5048	160	15	κ	κ	NOUN
ejpam-5048	160	16	,	,	PUNCT
ejpam-5048	160	17	δ	δ	PROPN
ejpam-5048	160	18	)	)	PUNCT
ejpam-5048	160	19	7→	7→	NOUN
ejpam-5048	160	20	−	−	PROPN
ejpam-5048	160	21	δ	δ	PROPN
ejpam-5048	160	22	κ	κ	PROPN
ejpam-5048	160	23	,	,	PUNCT
ejpam-5048	160	24	(	(	PUNCT
ejpam-5048	160	25	21	21	NUM
ejpam-5048	160	26	)	)	PUNCT
ejpam-5048	160	27	respectively	respectively	ADV
ejpam-5048	160	28	.	.	PUNCT
ejpam-5048	161	1	it	it	PRON
ejpam-5048	161	2	can	can	AUX
ejpam-5048	161	3	be	be	AUX
ejpam-5048	161	4	easily	easily	ADV
ejpam-5048	161	5	confirmed	confirm	VERB
ejpam-5048	161	6	that	that	SCONJ
ejpam-5048	161	7	(	(	PUNCT
ejpam-5048	161	8	x	x	X
ejpam-5048	161	9	,	,	PUNCT
ejpam-5048	161	10	∗+	∗+	PROPN
ejpam-5048	161	11	,	,	PUNCT
ejpam-5048	161	12	1	1	NUM
ejpam-5048	161	13	)	)	PUNCT
ejpam-5048	161	14	and	and	CCONJ
ejpam-5048	161	15	(	(	PUNCT
ejpam-5048	161	16	x	x	X
ejpam-5048	161	17	,	,	PUNCT
ejpam-5048	161	18	∗−,−1	∗−,−1	PRON
ejpam-5048	161	19	)	)	PUNCT
ejpam-5048	161	20	are	be	AUX
ejpam-5048	161	21	qge	qge	NOUN
ejpam-5048	161	22	-	-	PUNCT
ejpam-5048	161	23	algebras	algebras	X
ejpam-5048	161	24	.	.	PUNCT
ejpam-5048	162	1	let	let	VERB
ejpam-5048	163	1	e	e	NOUN
ejpam-5048	163	2	:	:	PUNCT
ejpam-5048	163	3	=	=	SYM
ejpam-5048	163	4	r+	r+	NOUN
ejpam-5048	163	5	and	and	CCONJ
ejpam-5048	163	6	d	d	NOUN
ejpam-5048	163	7	:	:	PUNCT
ejpam-5048	163	8	=	=	SYM
ejpam-5048	163	9	r−	r−	PROPN
ejpam-5048	163	10	be	be	VERB
ejpam-5048	163	11	the	the	DET
ejpam-5048	163	12	set	set	NOUN
ejpam-5048	163	13	of	of	ADP
ejpam-5048	163	14	all	all	DET
ejpam-5048	163	15	positive	positive	ADJ
ejpam-5048	163	16	real	real	ADJ
ejpam-5048	163	17	numbers	number	NOUN
ejpam-5048	163	18	and	and	CCONJ
ejpam-5048	163	19	the	the	DET
ejpam-5048	163	20	set	set	NOUN
ejpam-5048	163	21	of	of	ADP
ejpam-5048	163	22	all	all	DET
ejpam-5048	163	23	positive	positive	ADJ
ejpam-5048	163	24	real	real	ADJ
ejpam-5048	163	25	numbers	number	NOUN
ejpam-5048	163	26	,	,	PUNCT
ejpam-5048	163	27	respectively	respectively	ADV
ejpam-5048	163	28	.	.	PUNCT
ejpam-5048	164	1	then	then	ADV
ejpam-5048	164	2	e	e	PROPN
ejpam-5048	164	3	is	be	AUX
ejpam-5048	164	4	a	a	DET
ejpam-5048	164	5	qge	qge	NOUN
ejpam-5048	164	6	-	-	PUNCT
ejpam-5048	164	7	subalgebra	subalgebra	NOUN
ejpam-5048	164	8	of	of	ADP
ejpam-5048	164	9	(	(	PUNCT
ejpam-5048	164	10	x	x	PROPN
ejpam-5048	164	11	,	,	PUNCT
ejpam-5048	164	12	∗+	∗+	PROPN
ejpam-5048	164	13	,	,	PUNCT
ejpam-5048	164	14	1	1	NUM
ejpam-5048	164	15	)	)	PUNCT
ejpam-5048	164	16	,	,	PUNCT
ejpam-5048	164	17	but	but	CCONJ
ejpam-5048	164	18	d	d	NOUN
ejpam-5048	164	19	is	be	AUX
ejpam-5048	164	20	not	not	PART
ejpam-5048	164	21	a	a	DET
ejpam-5048	164	22	qge	qge	NOUN
ejpam-5048	164	23	-	-	PUNCT
ejpam-5048	164	24	subalgebra	subalgebra	NOUN
ejpam-5048	164	25	of	of	ADP
ejpam-5048	164	26	(	(	PUNCT
ejpam-5048	164	27	x	x	PROPN
ejpam-5048	164	28	,	,	PUNCT
ejpam-5048	164	29	∗+	∗+	PROPN
ejpam-5048	164	30	,	,	PUNCT
ejpam-5048	164	31	1	1	NUM
ejpam-5048	164	32	)	)	PUNCT
ejpam-5048	164	33	.	.	PUNCT
ejpam-5048	165	1	also	also	ADV
ejpam-5048	165	2	d	d	PRON
ejpam-5048	165	3	is	be	AUX
ejpam-5048	165	4	a	a	DET
ejpam-5048	165	5	qge	qge	NOUN
ejpam-5048	165	6	-	-	PUNCT
ejpam-5048	165	7	subalgebra	subalgebra	NOUN
ejpam-5048	165	8	of	of	ADP
ejpam-5048	165	9	(	(	PUNCT
ejpam-5048	165	10	x	x	X
ejpam-5048	165	11	,	,	PUNCT
ejpam-5048	165	12	∗−,−1	∗−,−1	NUM
ejpam-5048	165	13	)	)	PUNCT
ejpam-5048	165	14	,	,	PUNCT
ejpam-5048	165	15	but	but	CCONJ
ejpam-5048	165	16	e	e	NOUN
ejpam-5048	165	17	is	be	AUX
ejpam-5048	165	18	not	not	PART
ejpam-5048	165	19	a	a	DET
ejpam-5048	165	20	qge	qge	NOUN
ejpam-5048	165	21	-	-	PUNCT
ejpam-5048	165	22	subalgebra	subalgebra	NOUN
ejpam-5048	165	23	of	of	ADP
ejpam-5048	165	24	(	(	PUNCT
ejpam-5048	165	25	x	x	X
ejpam-5048	165	26	,	,	PUNCT
ejpam-5048	165	27	∗−,−1	∗−,−1	NUM
ejpam-5048	165	28	)	)	PUNCT
ejpam-5048	165	29	.	.	PUNCT
ejpam-5048	166	1	y.	y.	PROPN
ejpam-5048	166	2	b.	b.	PROPN
ejpam-5048	166	3	jun	jun	PROPN
ejpam-5048	166	4	,	,	PUNCT
ejpam-5048	166	5	ravikumar	ravikumar	PROPN
ejpam-5048	166	6	bandaru	bandaru	PROPN
ejpam-5048	166	7	,	,	PUNCT
ejpam-5048	166	8	rahul	rahul	PROPN
ejpam-5048	166	9	shukla	shukla	PROPN
ejpam-5048	166	10	/	/	SYM
ejpam-5048	166	11	eur	eur	PROPN
ejpam-5048	166	12	.	.	PUNCT
ejpam-5048	167	1	j.	j.	PROPN
ejpam-5048	167	2	pure	pure	PROPN
ejpam-5048	167	3	appl	appl	PROPN
ejpam-5048	167	4	.	.	PROPN
ejpam-5048	167	5	math	math	PROPN
ejpam-5048	167	6	,	,	PUNCT
ejpam-5048	167	7	17	17	NUM
ejpam-5048	167	8	(	(	PUNCT
ejpam-5048	167	9	1	1	NUM
ejpam-5048	167	10	)	)	PUNCT
ejpam-5048	167	11	(	(	PUNCT
ejpam-5048	167	12	2024	2024	NUM
ejpam-5048	167	13	)	)	PUNCT
ejpam-5048	167	14	,	,	PUNCT
ejpam-5048	167	15	569	569	NUM
ejpam-5048	167	16	-	-	SYM
ejpam-5048	167	17	581	581	NUM
ejpam-5048	167	18	575	575	NUM
ejpam-5048	167	19	proposition	proposition	NOUN
ejpam-5048	167	20	3	3	NUM
ejpam-5048	167	21	.	.	PUNCT
ejpam-5048	168	1	every	every	DET
ejpam-5048	168	2	qge	qge	NOUN
ejpam-5048	168	3	-	-	PUNCT
ejpam-5048	168	4	subalgebra	subalgebra	NOUN
ejpam-5048	168	5	of	of	ADP
ejpam-5048	168	6	x	x	PUNCT
ejpam-5048	168	7	contains	contain	VERB
ejpam-5048	168	8	the	the	DET
ejpam-5048	168	9	unit	unit	NOUN
ejpam-5048	168	10	1	1	NUM
ejpam-5048	168	11	.	.	PUNCT
ejpam-5048	169	1	proof	proof	NOUN
ejpam-5048	169	2	.	.	PUNCT
ejpam-5048	170	1	it	it	PRON
ejpam-5048	170	2	is	be	AUX
ejpam-5048	170	3	straightforward	straightforward	ADJ
ejpam-5048	170	4	by	by	ADP
ejpam-5048	170	5	(	(	PUNCT
ejpam-5048	170	6	ge1	ge1	NOUN
ejpam-5048	170	7	)	)	PUNCT
ejpam-5048	170	8	.	.	PUNCT
ejpam-5048	171	1	theorem	theorem	NOUN
ejpam-5048	171	2	3	3	X
ejpam-5048	171	3	.	.	PUNCT
ejpam-5048	172	1	let	let	AUX
ejpam-5048	172	2	(	(	PUNCT
ejpam-5048	172	3	z	z	NOUN
ejpam-5048	172	4	,	,	PUNCT
ejpam-5048	172	5	∗	∗	NOUN
ejpam-5048	172	6	,	,	PUNCT
ejpam-5048	172	7	1	1	NUM
ejpam-5048	172	8	)	)	PUNCT
ejpam-5048	172	9	be	be	AUX
ejpam-5048	172	10	the	the	DET
ejpam-5048	172	11	product	product	NOUN
ejpam-5048	172	12	qge	qge	NOUN
ejpam-5048	172	13	-	-	NOUN
ejpam-5048	172	14	algebra	algebra	NOUN
ejpam-5048	172	15	of	of	ADP
ejpam-5048	172	16	qge	qge	NOUN
ejpam-5048	172	17	-	-	PUNCT
ejpam-5048	172	18	algebras	algebras	X
ejpam-5048	172	19	(	(	PUNCT
ejpam-5048	172	20	x	x	X
ejpam-5048	172	21	,	,	PUNCT
ejpam-5048	172	22	∗x	∗x	PROPN
ejpam-5048	172	23	,	,	PUNCT
ejpam-5048	172	24	1x	1x	NUM
ejpam-5048	172	25	)	)	PUNCT
ejpam-5048	172	26	and	and	CCONJ
ejpam-5048	172	27	(	(	PUNCT
ejpam-5048	172	28	y	y	PROPN
ejpam-5048	172	29	,	,	PUNCT
ejpam-5048	172	30	∗y	∗y	PROPN
ejpam-5048	172	31	,	,	PUNCT
ejpam-5048	172	32	1y	1y	PROPN
ejpam-5048	172	33	)	)	PUNCT
ejpam-5048	172	34	.	.	PUNCT
ejpam-5048	173	1	if	if	SCONJ
ejpam-5048	173	2	d	d	PROPN
ejpam-5048	173	3	and	and	CCONJ
ejpam-5048	173	4	e	e	PROPN
ejpam-5048	173	5	are	be	AUX
ejpam-5048	173	6	qge	qge	NOUN
ejpam-5048	173	7	-	-	PUNCT
ejpam-5048	173	8	subalgebras	subalgebra	NOUN
ejpam-5048	173	9	of	of	ADP
ejpam-5048	173	10	x	x	PROPN
ejpam-5048	173	11	and	and	CCONJ
ejpam-5048	173	12	y	y	PROPN
ejpam-5048	173	13	,	,	PUNCT
ejpam-5048	173	14	respectively	respectively	ADV
ejpam-5048	173	15	,	,	PUNCT
ejpam-5048	173	16	then	then	ADV
ejpam-5048	173	17	their	their	PRON
ejpam-5048	173	18	product	product	NOUN
ejpam-5048	173	19	d	d	X
ejpam-5048	173	20	×	×	NOUN
ejpam-5048	173	21	e	e	NOUN
ejpam-5048	173	22	is	be	AUX
ejpam-5048	173	23	a	a	DET
ejpam-5048	173	24	qge	qge	NOUN
ejpam-5048	173	25	-	-	PUNCT
ejpam-5048	173	26	subalgebra	subalgebra	NOUN
ejpam-5048	173	27	of	of	ADP
ejpam-5048	173	28	z.	z.	PROPN
ejpam-5048	173	29	proof	proof	PROPN
ejpam-5048	173	30	.	.	PUNCT
ejpam-5048	174	1	let	let	VERB
ejpam-5048	174	2	(	(	PUNCT
ejpam-5048	174	3	κ	κ	NOUN
ejpam-5048	174	4	,	,	PUNCT
ejpam-5048	174	5	δ	δ	PROPN
ejpam-5048	174	6	)	)	PUNCT
ejpam-5048	174	7	,	,	PUNCT
ejpam-5048	174	8	(	(	PUNCT
ejpam-5048	174	9	ϖ,π	ϖ,π	ADJ
ejpam-5048	174	10	)	)	PUNCT
ejpam-5048	174	11	∈	∈	PROPN
ejpam-5048	175	1	d	d	X
ejpam-5048	175	2	×	×	PROPN
ejpam-5048	175	3	e.	e.	PROPN
ejpam-5048	175	4	then	then	ADV
ejpam-5048	175	5	κ,ϖ	κ,ϖ	PROPN
ejpam-5048	175	6	∈	∈	PROPN
ejpam-5048	175	7	d	d	PROPN
ejpam-5048	175	8	and	and	CCONJ
ejpam-5048	175	9	δ	δ	PROPN
ejpam-5048	175	10	,	,	PUNCT
ejpam-5048	175	11	π	π	PROPN
ejpam-5048	175	12	∈	∈	PROPN
ejpam-5048	175	13	e	e	NOUN
ejpam-5048	175	14	,	,	PUNCT
ejpam-5048	175	15	and	and	CCONJ
ejpam-5048	175	16	thus	thus	ADV
ejpam-5048	175	17	κ	κ	ADP
ejpam-5048	175	18	∗x	∗x	PROPN
ejpam-5048	175	19	ϖ	ϖ	X
ejpam-5048	175	20	∈	∈	PROPN
ejpam-5048	175	21	d	d	NOUN
ejpam-5048	175	22	and	and	CCONJ
ejpam-5048	175	23	δ	δ	PROPN
ejpam-5048	175	24	∗y	∗y	PROPN
ejpam-5048	175	25	π	π	PROPN
ejpam-5048	175	26	∈	∈	PROPN
ejpam-5048	176	1	e.	e.	PROPN
ejpam-5048	177	1	it	it	PRON
ejpam-5048	177	2	follows	follow	VERB
ejpam-5048	177	3	that	that	SCONJ
ejpam-5048	177	4	(	(	PUNCT
ejpam-5048	177	5	κ	κ	NOUN
ejpam-5048	177	6	,	,	PUNCT
ejpam-5048	177	7	δ	δ	PROPN
ejpam-5048	177	8	)	)	PUNCT
ejpam-5048	177	9	∗	∗	NOUN
ejpam-5048	177	10	(	(	PUNCT
ejpam-5048	177	11	ϖ,π	ϖ,π	PROPN
ejpam-5048	177	12	)	)	PUNCT
ejpam-5048	177	13	=	=	SYM
ejpam-5048	178	1	(	(	PUNCT
ejpam-5048	178	2	κ	κ	X
ejpam-5048	178	3	∗x	∗x	PROPN
ejpam-5048	178	4	δ,ϖ	δ,ϖ	PROPN
ejpam-5048	178	5	∗y	∗y	PROPN
ejpam-5048	178	6	π	π	PROPN
ejpam-5048	178	7	)	)	PUNCT
ejpam-5048	178	8	∈	∈	PROPN
ejpam-5048	179	1	d	d	X
ejpam-5048	179	2	×	×	PROPN
ejpam-5048	179	3	e.	e.	PROPN
ejpam-5048	180	1	hence	hence	ADV
ejpam-5048	180	2	d	d	PROPN
ejpam-5048	180	3	×	×	PROPN
ejpam-5048	180	4	e	e	NOUN
ejpam-5048	180	5	is	be	AUX
ejpam-5048	180	6	a	a	DET
ejpam-5048	180	7	qge	qge	NOUN
ejpam-5048	180	8	-	-	PUNCT
ejpam-5048	180	9	subalgebra	subalgebra	NOUN
ejpam-5048	180	10	of	of	ADP
ejpam-5048	180	11	z.	z.	PROPN
ejpam-5048	180	12	the	the	DET
ejpam-5048	180	13	following	follow	VERB
ejpam-5048	180	14	example	example	NOUN
ejpam-5048	180	15	illustrates	illustrate	VERB
ejpam-5048	180	16	theorem	theorem	VERB
ejpam-5048	180	17	3	3	NUM
ejpam-5048	180	18	.	.	NOUN
ejpam-5048	180	19	example	example	NOUN
ejpam-5048	180	20	11	11	NUM
ejpam-5048	180	21	.	.	PUNCT
ejpam-5048	181	1	consider	consider	VERB
ejpam-5048	181	2	the	the	DET
ejpam-5048	181	3	qge	qge	NOUN
ejpam-5048	181	4	-	-	NOUN
ejpam-5048	181	5	algebra	algebra	NOUN
ejpam-5048	181	6	(	(	PUNCT
ejpam-5048	181	7	x	x	X
ejpam-5048	181	8	,	,	PUNCT
ejpam-5048	181	9	∗x	∗x	PROPN
ejpam-5048	181	10	,	,	PUNCT
ejpam-5048	181	11	1x	1x	NUM
ejpam-5048	181	12	)	)	PUNCT
ejpam-5048	181	13	given	give	VERB
ejpam-5048	181	14	in	in	ADP
ejpam-5048	181	15	example	example	NOUN
ejpam-5048	181	16	2	2	NUM
ejpam-5048	181	17	and	and	CCONJ
ejpam-5048	181	18	the	the	DET
ejpam-5048	181	19	qgealgebra	qgealgebra	NOUN
ejpam-5048	181	20	(	(	PUNCT
ejpam-5048	181	21	r,−	r,−	PROPN
ejpam-5048	181	22	,	,	PUNCT
ejpam-5048	181	23	0	0	NUM
ejpam-5048	181	24	)	)	PUNCT
ejpam-5048	181	25	which	which	PRON
ejpam-5048	181	26	is	be	AUX
ejpam-5048	181	27	given	give	VERB
ejpam-5048	181	28	in	in	ADP
ejpam-5048	181	29	example	example	NOUN
ejpam-5048	181	30	4	4	NUM
ejpam-5048	181	31	.	.	PUNCT
ejpam-5048	182	1	then	then	ADV
ejpam-5048	182	2	(	(	PUNCT
ejpam-5048	182	3	x×r	x×r	PROPN
ejpam-5048	182	4	,	,	PUNCT
ejpam-5048	182	5	∗	∗	NOUN
ejpam-5048	182	6	,	,	PUNCT
ejpam-5048	182	7	1	1	NUM
ejpam-5048	182	8	)	)	PUNCT
ejpam-5048	182	9	is	be	AUX
ejpam-5048	182	10	the	the	DET
ejpam-5048	182	11	product	product	NOUN
ejpam-5048	182	12	qge	qge	NOUN
ejpam-5048	182	13	-	-	NOUN
ejpam-5048	182	14	algebra	algebra	NOUN
ejpam-5048	182	15	of	of	ADP
ejpam-5048	182	16	(	(	PUNCT
ejpam-5048	182	17	x	x	NOUN
ejpam-5048	182	18	,	,	PUNCT
ejpam-5048	182	19	∗x	∗x	NOUN
ejpam-5048	182	20	,	,	PUNCT
ejpam-5048	182	21	1x	1x	NUM
ejpam-5048	182	22	)	)	PUNCT
ejpam-5048	182	23	and	and	CCONJ
ejpam-5048	182	24	(	(	PUNCT
ejpam-5048	182	25	r,−	r,−	PROPN
ejpam-5048	182	26	,	,	PUNCT
ejpam-5048	182	27	0	0	NUM
ejpam-5048	182	28	)	)	PUNCT
ejpam-5048	182	29	where	where	SCONJ
ejpam-5048	182	30	∗	∗	NOUN
ejpam-5048	182	31	is	be	AUX
ejpam-5048	182	32	defined	define	VERB
ejpam-5048	182	33	by	by	ADP
ejpam-5048	182	34	(	(	PUNCT
ejpam-5048	182	35	∀(κ	∀(κ	PROPN
ejpam-5048	182	36	,	,	PUNCT
ejpam-5048	182	37	δ	δ	PROPN
ejpam-5048	182	38	)	)	PUNCT
ejpam-5048	182	39	,	,	PUNCT
ejpam-5048	182	40	(	(	PUNCT
ejpam-5048	182	41	r	r	NOUN
ejpam-5048	182	42	,	,	PUNCT
ejpam-5048	182	43	s	s	NOUN
ejpam-5048	182	44	)	)	PUNCT
ejpam-5048	182	45	∈	∈	PROPN
ejpam-5048	182	46	x	x	X
ejpam-5048	182	47	×	×	PROPN
ejpam-5048	182	48	r)((κ	r)((κ	PRON
ejpam-5048	182	49	,	,	PUNCT
ejpam-5048	182	50	δ	δ	PROPN
ejpam-5048	182	51	)	)	PUNCT
ejpam-5048	182	52	∗	∗	NOUN
ejpam-5048	182	53	(	(	PUNCT
ejpam-5048	182	54	r	r	NOUN
ejpam-5048	182	55	,	,	PUNCT
ejpam-5048	182	56	s	s	NOUN
ejpam-5048	182	57	)	)	PUNCT
ejpam-5048	182	58	=	=	SYM
ejpam-5048	182	59	(	(	PUNCT
ejpam-5048	182	60	κ	κ	X
ejpam-5048	182	61	∗x	∗x	PROPN
ejpam-5048	182	62	r	r	NOUN
ejpam-5048	182	63	,	,	PUNCT
ejpam-5048	182	64	s−	s−	PROPN
ejpam-5048	182	65	δ	δ	PROPN
ejpam-5048	182	66	)	)	PUNCT
ejpam-5048	182	67	)	)	PUNCT
ejpam-5048	182	68	.	.	PUNCT
ejpam-5048	183	1	let	let	VERB
ejpam-5048	183	2	d	d	NOUN
ejpam-5048	183	3	=	=	PUNCT
ejpam-5048	183	4	{	{	PUNCT
ejpam-5048	183	5	1	1	NUM
ejpam-5048	183	6	,	,	PUNCT
ejpam-5048	183	7	a	a	PRON
ejpam-5048	183	8	}	}	PUNCT
ejpam-5048	183	9	and	and	CCONJ
ejpam-5048	183	10	e	e	X
ejpam-5048	183	11	=	=	PROPN
ejpam-5048	183	12	z.	z.	PROPN
ejpam-5048	184	1	then	then	ADV
ejpam-5048	184	2	d	d	PROPN
ejpam-5048	184	3	and	and	CCONJ
ejpam-5048	184	4	e	e	PROPN
ejpam-5048	184	5	are	be	AUX
ejpam-5048	184	6	qge	qge	NOUN
ejpam-5048	184	7	-	-	PUNCT
ejpam-5048	184	8	subalgebras	subalgebra	NOUN
ejpam-5048	184	9	of	of	ADP
ejpam-5048	184	10	x	x	PROPN
ejpam-5048	184	11	and	and	CCONJ
ejpam-5048	184	12	r	r	NOUN
ejpam-5048	184	13	,	,	PUNCT
ejpam-5048	184	14	respectively	respectively	ADV
ejpam-5048	184	15	.	.	PUNCT
ejpam-5048	185	1	let	let	VERB
ejpam-5048	185	2	(	(	PUNCT
ejpam-5048	185	3	κ	κ	NOUN
ejpam-5048	185	4	,	,	PUNCT
ejpam-5048	185	5	δ	δ	PROPN
ejpam-5048	185	6	)	)	PUNCT
ejpam-5048	185	7	,	,	PUNCT
ejpam-5048	185	8	(	(	PUNCT
ejpam-5048	185	9	u	u	NOUN
ejpam-5048	185	10	,	,	PUNCT
ejpam-5048	185	11	v	v	NOUN
ejpam-5048	185	12	)	)	PUNCT
ejpam-5048	185	13	∈	∈	PROPN
ejpam-5048	185	14	d×e	d×e	PROPN
ejpam-5048	185	15	.	.	PUNCT
ejpam-5048	186	1	then	then	ADV
ejpam-5048	186	2	κ	κ	X
ejpam-5048	186	3	,	,	PUNCT
ejpam-5048	186	4	u	u	PROPN
ejpam-5048	186	5	∈	∈	PROPN
ejpam-5048	186	6	d	d	PROPN
ejpam-5048	186	7	and	and	CCONJ
ejpam-5048	186	8	δ	δ	PROPN
ejpam-5048	186	9	,	,	PUNCT
ejpam-5048	186	10	v	v	ADP
ejpam-5048	186	11	∈	∈	NOUN
ejpam-5048	186	12	e	e	NOUN
ejpam-5048	186	13	,	,	PUNCT
ejpam-5048	186	14	and	and	CCONJ
ejpam-5048	186	15	thus	thus	ADV
ejpam-5048	186	16	κ∗x	κ∗x	NUM
ejpam-5048	186	17	u	u	NOUN
ejpam-5048	186	18	∈	∈	PROPN
ejpam-5048	186	19	d	d	NOUN
ejpam-5048	186	20	and	and	CCONJ
ejpam-5048	186	21	v−δ	v−δ	PROPN
ejpam-5048	186	22	∈	∈	PROPN
ejpam-5048	186	23	e.	e.	PROPN
ejpam-5048	187	1	it	it	PRON
ejpam-5048	187	2	follows	follow	VERB
ejpam-5048	187	3	that	that	SCONJ
ejpam-5048	187	4	(	(	PUNCT
ejpam-5048	187	5	κ	κ	NOUN
ejpam-5048	187	6	,	,	PUNCT
ejpam-5048	187	7	δ	δ	PROPN
ejpam-5048	187	8	)	)	PUNCT
ejpam-5048	187	9	∗	∗	NOUN
ejpam-5048	187	10	(	(	PUNCT
ejpam-5048	187	11	u	u	NOUN
ejpam-5048	187	12	,	,	PUNCT
ejpam-5048	187	13	v	v	NOUN
ejpam-5048	187	14	)	)	PUNCT
ejpam-5048	187	15	=	=	SYM
ejpam-5048	187	16	(	(	PUNCT
ejpam-5048	187	17	κ	κ	X
ejpam-5048	187	18	∗x	∗x	PROPN
ejpam-5048	187	19	u	u	NOUN
ejpam-5048	187	20	,	,	PUNCT
ejpam-5048	187	21	v	v	ADP
ejpam-5048	187	22	−	−	PROPN
ejpam-5048	187	23	δ	δ	PROPN
ejpam-5048	187	24	)	)	PUNCT
ejpam-5048	187	25	∈	∈	PROPN
ejpam-5048	188	1	d	d	X
ejpam-5048	188	2	×	×	PROPN
ejpam-5048	188	3	e.	e.	PROPN
ejpam-5048	189	1	hence	hence	ADV
ejpam-5048	189	2	d	d	PROPN
ejpam-5048	189	3	×	×	PROPN
ejpam-5048	189	4	e	e	NOUN
ejpam-5048	189	5	is	be	AUX
ejpam-5048	189	6	a	a	DET
ejpam-5048	189	7	qge	qge	NOUN
ejpam-5048	189	8	-	-	PUNCT
ejpam-5048	189	9	subalgebra	subalgebra	NOUN
ejpam-5048	189	10	of	of	ADP
ejpam-5048	189	11	x	x	SYM
ejpam-5048	189	12	×	×	PROPN
ejpam-5048	189	13	r.	r.	PROPN
ejpam-5048	189	14	theorem	theorem	VERB
ejpam-5048	189	15	4	4	NUM
ejpam-5048	189	16	.	.	PUNCT
ejpam-5048	190	1	the	the	DET
ejpam-5048	190	2	intersection	intersection	NOUN
ejpam-5048	190	3	of	of	ADP
ejpam-5048	190	4	two	two	NUM
ejpam-5048	190	5	qge	qge	NOUN
ejpam-5048	190	6	-	-	PUNCT
ejpam-5048	190	7	subalgebras	subalgebras	PROPN
ejpam-5048	190	8	is	be	AUX
ejpam-5048	190	9	a	a	DET
ejpam-5048	190	10	qge	qge	NOUN
ejpam-5048	190	11	-	-	PUNCT
ejpam-5048	190	12	subalgebra	subalgebra	NOUN
ejpam-5048	190	13	.	.	PUNCT
ejpam-5048	191	1	the	the	DET
ejpam-5048	191	2	union	union	NOUN
ejpam-5048	191	3	of	of	ADP
ejpam-5048	191	4	two	two	NUM
ejpam-5048	191	5	qge	qge	NOUN
ejpam-5048	191	6	-	-	PUNCT
ejpam-5048	191	7	subalgebras	subalgebras	PROPN
ejpam-5048	191	8	may	may	AUX
ejpam-5048	191	9	not	not	PART
ejpam-5048	191	10	be	be	AUX
ejpam-5048	191	11	a	a	DET
ejpam-5048	191	12	qge	qge	NOUN
ejpam-5048	191	13	-	-	PUNCT
ejpam-5048	191	14	subalgebra	subalgebra	NOUN
ejpam-5048	191	15	as	as	SCONJ
ejpam-5048	191	16	shown	show	VERB
ejpam-5048	191	17	in	in	ADP
ejpam-5048	191	18	the	the	DET
ejpam-5048	191	19	following	follow	VERB
ejpam-5048	191	20	example	example	NOUN
ejpam-5048	191	21	.	.	PUNCT
ejpam-5048	192	1	example	example	NOUN
ejpam-5048	192	2	12	12	NUM
ejpam-5048	192	3	.	.	PUNCT
ejpam-5048	193	1	consider	consider	VERB
ejpam-5048	193	2	the	the	DET
ejpam-5048	193	3	qge	qge	NOUN
ejpam-5048	193	4	-	-	NOUN
ejpam-5048	193	5	algebra	algebra	NOUN
ejpam-5048	193	6	x	x	PUNCT
ejpam-5048	193	7	given	give	VERB
ejpam-5048	193	8	in	in	ADP
ejpam-5048	193	9	example	example	NOUN
ejpam-5048	193	10	2	2	X
ejpam-5048	193	11	.	.	PUNCT
ejpam-5048	194	1	it	it	PRON
ejpam-5048	194	2	is	be	AUX
ejpam-5048	194	3	routine	routine	ADJ
ejpam-5048	194	4	to	to	PART
ejpam-5048	194	5	verify	verify	VERB
ejpam-5048	194	6	that	that	SCONJ
ejpam-5048	194	7	the	the	DET
ejpam-5048	194	8	set	set	NOUN
ejpam-5048	194	9	e1	e1	NOUN
ejpam-5048	194	10	=	=	SYM
ejpam-5048	194	11	{	{	PUNCT
ejpam-5048	194	12	1	1	NUM
ejpam-5048	194	13	,	,	PUNCT
ejpam-5048	194	14	a	a	PRON
ejpam-5048	194	15	}	}	PUNCT
ejpam-5048	194	16	and	and	CCONJ
ejpam-5048	194	17	e2	e2	PROPN
ejpam-5048	194	18	=	=	PUNCT
ejpam-5048	194	19	{	{	PUNCT
ejpam-5048	194	20	1	1	NUM
ejpam-5048	194	21	,	,	PUNCT
ejpam-5048	194	22	c	c	NOUN
ejpam-5048	194	23	}	}	PUNCT
ejpam-5048	194	24	are	be	AUX
ejpam-5048	194	25	qge	qge	NOUN
ejpam-5048	194	26	-	-	PUNCT
ejpam-5048	194	27	subalgebras	subalgebras	PROPN
ejpam-5048	194	28	of	of	ADP
ejpam-5048	194	29	x.	x.	PROPN
ejpam-5048	194	30	but	but	CCONJ
ejpam-5048	194	31	e1∪e2	e1∪e2	X
ejpam-5048	194	32	=	=	PUNCT
ejpam-5048	194	33	{	{	PUNCT
ejpam-5048	194	34	1	1	NUM
ejpam-5048	194	35	,	,	PUNCT
ejpam-5048	194	36	a	a	DET
ejpam-5048	194	37	,	,	PUNCT
ejpam-5048	194	38	c	c	NOUN
ejpam-5048	194	39	}	}	PUNCT
ejpam-5048	194	40	is	be	AUX
ejpam-5048	194	41	not	not	PART
ejpam-5048	194	42	a	a	DET
ejpam-5048	194	43	qge	qge	NOUN
ejpam-5048	194	44	-	-	PUNCT
ejpam-5048	194	45	subalgebra	subalgebra	NOUN
ejpam-5048	194	46	of	of	ADP
ejpam-5048	194	47	x	x	PRON
ejpam-5048	194	48	since	since	SCONJ
ejpam-5048	194	49	a	a	PRON
ejpam-5048	194	50	,	,	PUNCT
ejpam-5048	194	51	c	c	PROPN
ejpam-5048	194	52	∈	∈	PROPN
ejpam-5048	194	53	e1	e1	PROPN
ejpam-5048	194	54	∪	∪	NOUN
ejpam-5048	194	55	e2	e2	PROPN
ejpam-5048	194	56	but	but	CCONJ
ejpam-5048	194	57	a	a	DET
ejpam-5048	194	58	∗	∗	NOUN
ejpam-5048	194	59	c	c	NOUN
ejpam-5048	194	60	=	=	SYM
ejpam-5048	194	61	b	b	PROPN
ejpam-5048	194	62	/∈	/∈	PUNCT
ejpam-5048	194	63	e1	e1	PROPN
ejpam-5048	194	64	∪	∪	PROPN
ejpam-5048	194	65	e2	e2	PROPN
ejpam-5048	194	66	.	.	PUNCT
ejpam-5048	195	1	5	5	NUM
ejpam-5048	195	2	.	.	X
ejpam-5048	195	3	qge	qge	NOUN
ejpam-5048	195	4	-	-	NOUN
ejpam-5048	195	5	filters	filter	NOUN
ejpam-5048	195	6	in	in	ADP
ejpam-5048	195	7	this	this	DET
ejpam-5048	195	8	section	section	NOUN
ejpam-5048	195	9	,	,	PUNCT
ejpam-5048	195	10	we	we	PRON
ejpam-5048	195	11	introduce	introduce	VERB
ejpam-5048	195	12	the	the	DET
ejpam-5048	195	13	qge	qge	NOUN
ejpam-5048	195	14	-	-	NOUN
ejpam-5048	195	15	filter	filter	NOUN
ejpam-5048	195	16	in	in	ADP
ejpam-5048	195	17	a	a	DET
ejpam-5048	195	18	qge	qge	NOUN
ejpam-5048	195	19	-	-	NOUN
ejpam-5048	195	20	algebra	algebra	NOUN
ejpam-5048	195	21	in	in	ADP
ejpam-5048	195	22	the	the	DET
ejpam-5048	195	23	same	same	ADJ
ejpam-5048	195	24	way	way	NOUN
ejpam-5048	195	25	as	as	SCONJ
ejpam-5048	195	26	the	the	DET
ejpam-5048	195	27	ge	ge	NOUN
ejpam-5048	195	28	-	-	NOUN
ejpam-5048	195	29	filter	filter	NOUN
ejpam-5048	195	30	in	in	ADP
ejpam-5048	195	31	a	a	DET
ejpam-5048	195	32	ge	ge	NOUN
ejpam-5048	195	33	-	-	NOUN
ejpam-5048	195	34	algebra	algebra	PROPN
ejpam-5048	195	35	as	as	SCONJ
ejpam-5048	195	36	follows	follow	VERB
ejpam-5048	195	37	.	.	PUNCT
ejpam-5048	196	1	definition	definition	NOUN
ejpam-5048	196	2	3	3	NUM
ejpam-5048	196	3	.	.	PUNCT
ejpam-5048	197	1	a	a	DET
ejpam-5048	197	2	subset	subset	NOUN
ejpam-5048	197	3	f	f	NOUN
ejpam-5048	197	4	of	of	ADP
ejpam-5048	197	5	x	x	PROPN
ejpam-5048	197	6	is	be	AUX
ejpam-5048	197	7	called	call	VERB
ejpam-5048	197	8	a	a	DET
ejpam-5048	197	9	qge	qge	NOUN
ejpam-5048	197	10	-	-	NOUN
ejpam-5048	197	11	filter	filter	NOUN
ejpam-5048	197	12	of	of	ADP
ejpam-5048	197	13	x	x	PRON
ejpam-5048	197	14	if	if	SCONJ
ejpam-5048	197	15	it	it	PRON
ejpam-5048	197	16	satisfies	satisfy	VERB
ejpam-5048	197	17	:	:	PUNCT
ejpam-5048	197	18	1	1	NUM
ejpam-5048	197	19	∈	∈	NOUN
ejpam-5048	197	20	f	f	X
ejpam-5048	197	21	,	,	PUNCT
ejpam-5048	197	22	(	(	PUNCT
ejpam-5048	197	23	22	22	NUM
ejpam-5048	197	24	)	)	PUNCT
ejpam-5048	197	25	(	(	PUNCT
ejpam-5048	197	26	∀κ	∀κ	ADV
ejpam-5048	197	27	,	,	PUNCT
ejpam-5048	197	28	δ	δ	PROPN
ejpam-5048	197	29	∈	∈	PROPN
ejpam-5048	197	30	x)(κ	x)(κ	PROPN
ejpam-5048	197	31	∗	∗	VERB
ejpam-5048	197	32	δ	δ	PROPN
ejpam-5048	197	33	∈	∈	PROPN
ejpam-5048	197	34	f	f	PROPN
ejpam-5048	197	35	,	,	PUNCT
ejpam-5048	197	36	κ	κ	PROPN
ejpam-5048	197	37	∈	∈	PROPN
ejpam-5048	197	38	f	f	PROPN
ejpam-5048	197	39	⇒	⇒	VERB
ejpam-5048	197	40	δ	δ	PROPN
ejpam-5048	197	41	∈	∈	PROPN
ejpam-5048	197	42	f	f	PROPN
ejpam-5048	197	43	)	)	PUNCT
ejpam-5048	197	44	.	.	PUNCT
ejpam-5048	198	1	(	(	PUNCT
ejpam-5048	198	2	23	23	X
ejpam-5048	198	3	)	)	PUNCT
ejpam-5048	198	4	y.	y.	PROPN
ejpam-5048	198	5	b.	b.	PROPN
ejpam-5048	198	6	jun	jun	PROPN
ejpam-5048	198	7	,	,	PUNCT
ejpam-5048	198	8	ravikumar	ravikumar	PROPN
ejpam-5048	198	9	bandaru	bandaru	PROPN
ejpam-5048	198	10	,	,	PUNCT
ejpam-5048	198	11	rahul	rahul	PROPN
ejpam-5048	198	12	shukla	shukla	PROPN
ejpam-5048	198	13	/	/	SYM
ejpam-5048	198	14	eur	eur	PROPN
ejpam-5048	198	15	.	.	PUNCT
ejpam-5048	199	1	j.	j.	PROPN
ejpam-5048	199	2	pure	pure	PROPN
ejpam-5048	199	3	appl	appl	PROPN
ejpam-5048	199	4	.	.	PROPN
ejpam-5048	199	5	math	math	PROPN
ejpam-5048	199	6	,	,	PUNCT
ejpam-5048	199	7	17	17	NUM
ejpam-5048	199	8	(	(	PUNCT
ejpam-5048	199	9	1	1	NUM
ejpam-5048	199	10	)	)	PUNCT
ejpam-5048	199	11	(	(	PUNCT
ejpam-5048	199	12	2024	2024	NUM
ejpam-5048	199	13	)	)	PUNCT
ejpam-5048	199	14	,	,	PUNCT
ejpam-5048	199	15	569	569	NUM
ejpam-5048	199	16	-	-	SYM
ejpam-5048	199	17	581	581	NUM
ejpam-5048	199	18	576	576	NUM
ejpam-5048	199	19	example	example	NOUN
ejpam-5048	199	20	13	13	NUM
ejpam-5048	199	21	.	.	PUNCT
ejpam-5048	200	1	let	let	VERB
ejpam-5048	200	2	x	x	PUNCT
ejpam-5048	200	3	=	=	PRON
ejpam-5048	200	4	{	{	PUNCT
ejpam-5048	200	5	1	1	NUM
ejpam-5048	200	6	,	,	PUNCT
ejpam-5048	200	7	a	a	DET
ejpam-5048	200	8	,	,	PUNCT
ejpam-5048	200	9	b	b	NOUN
ejpam-5048	200	10	,	,	PUNCT
ejpam-5048	200	11	c	c	NOUN
ejpam-5048	200	12	,	,	PUNCT
ejpam-5048	200	13	d	d	NOUN
ejpam-5048	200	14	,	,	PUNCT
ejpam-5048	200	15	e	e	AUX
ejpam-5048	200	16	}	}	PUNCT
ejpam-5048	200	17	be	be	AUX
ejpam-5048	200	18	a	a	DET
ejpam-5048	200	19	set	set	NOUN
ejpam-5048	200	20	with	with	ADP
ejpam-5048	200	21	a	a	DET
ejpam-5048	200	22	binary	binary	ADJ
ejpam-5048	200	23	operation	operation	NOUN
ejpam-5048	200	24	“	"	PUNCT
ejpam-5048	200	25	∗	∗	NOUN
ejpam-5048	200	26	”	"	PUNCT
ejpam-5048	200	27	given	give	VERB
ejpam-5048	200	28	in	in	ADP
ejpam-5048	200	29	the	the	DET
ejpam-5048	200	30	following	follow	VERB
ejpam-5048	200	31	table	table	NOUN
ejpam-5048	200	32	:	:	PUNCT
ejpam-5048	200	33	∗	∗	NOUN
ejpam-5048	200	34	1	1	NUM
ejpam-5048	200	35	a	a	DET
ejpam-5048	200	36	b	b	NOUN
ejpam-5048	200	37	c	c	NOUN
ejpam-5048	200	38	d	d	X
ejpam-5048	200	39	e	e	PROPN
ejpam-5048	200	40	1	1	NUM
ejpam-5048	200	41	1	1	NUM
ejpam-5048	200	42	a	a	DET
ejpam-5048	200	43	b	b	NOUN
ejpam-5048	200	44	c	c	NOUN
ejpam-5048	200	45	d	d	PROPN
ejpam-5048	200	46	e	e	PROPN
ejpam-5048	200	47	a	a	PRON
ejpam-5048	201	1	a	a	DET
ejpam-5048	201	2	1	1	NUM
ejpam-5048	201	3	c	c	NOUN
ejpam-5048	201	4	b	b	PROPN
ejpam-5048	201	5	e	e	PROPN
ejpam-5048	201	6	d	d	PROPN
ejpam-5048	201	7	b	b	PROPN
ejpam-5048	201	8	b	b	PROPN
ejpam-5048	201	9	d	d	PROPN
ejpam-5048	201	10	1	1	NUM
ejpam-5048	201	11	e	e	NOUN
ejpam-5048	201	12	a	a	PROPN
ejpam-5048	201	13	c	c	NOUN
ejpam-5048	201	14	c	c	NOUN
ejpam-5048	201	15	d	d	X
ejpam-5048	201	16	b	b	PROPN
ejpam-5048	201	17	e	e	ADP
ejpam-5048	201	18	1	1	NUM
ejpam-5048	201	19	c	c	NOUN
ejpam-5048	201	20	a	a	PRON
ejpam-5048	201	21	d	d	X
ejpam-5048	201	22	c	c	NOUN
ejpam-5048	201	23	e	e	NOUN
ejpam-5048	201	24	a	a	PROPN
ejpam-5048	201	25	d	d	PROPN
ejpam-5048	201	26	1	1	NUM
ejpam-5048	201	27	b	b	PROPN
ejpam-5048	201	28	e	e	X
ejpam-5048	201	29	e	e	X
ejpam-5048	201	30	c	c	PROPN
ejpam-5048	201	31	d	d	X
ejpam-5048	201	32	a	a	PRON
ejpam-5048	201	33	b	b	NOUN
ejpam-5048	201	34	1	1	NUM
ejpam-5048	201	35	then	then	ADV
ejpam-5048	201	36	(	(	PUNCT
ejpam-5048	201	37	x	x	X
ejpam-5048	201	38	,	,	PUNCT
ejpam-5048	201	39	∗	∗	NOUN
ejpam-5048	201	40	,	,	PUNCT
ejpam-5048	201	41	1	1	NUM
ejpam-5048	201	42	)	)	PUNCT
ejpam-5048	201	43	is	be	AUX
ejpam-5048	201	44	a	a	DET
ejpam-5048	201	45	qge	qge	NOUN
ejpam-5048	201	46	-	-	NOUN
ejpam-5048	201	47	algebra	algebra	NOUN
ejpam-5048	201	48	,	,	PUNCT
ejpam-5048	201	49	and	and	CCONJ
ejpam-5048	201	50	it	it	PRON
ejpam-5048	201	51	is	be	AUX
ejpam-5048	201	52	routine	routine	ADJ
ejpam-5048	201	53	to	to	PART
ejpam-5048	201	54	check	check	VERB
ejpam-5048	201	55	that	that	SCONJ
ejpam-5048	201	56	the	the	DET
ejpam-5048	201	57	set	set	NOUN
ejpam-5048	201	58	f	f	X
ejpam-5048	201	59	=	=	PUNCT
ejpam-5048	201	60	{	{	PUNCT
ejpam-5048	201	61	1	1	NUM
ejpam-5048	201	62	,	,	PUNCT
ejpam-5048	201	63	c	c	X
ejpam-5048	201	64	,	,	PUNCT
ejpam-5048	201	65	d	d	X
ejpam-5048	201	66	}	}	PUNCT
ejpam-5048	201	67	is	be	AUX
ejpam-5048	201	68	a	a	DET
ejpam-5048	201	69	qge	qge	NOUN
ejpam-5048	201	70	-	-	NOUN
ejpam-5048	201	71	filter	filter	NOUN
ejpam-5048	201	72	of	of	ADP
ejpam-5048	201	73	x.	x.	PROPN
ejpam-5048	201	74	example	example	NOUN
ejpam-5048	201	75	14	14	NUM
ejpam-5048	201	76	.	.	PUNCT
ejpam-5048	202	1	consider	consider	VERB
ejpam-5048	202	2	the	the	DET
ejpam-5048	202	3	qge	qge	NOUN
ejpam-5048	202	4	-	-	NOUN
ejpam-5048	202	5	algebra	algebra	PROPN
ejpam-5048	202	6	(	(	PUNCT
ejpam-5048	202	7	y	y	PROPN
ejpam-5048	202	8	,	,	PUNCT
ejpam-5048	202	9	∗	∗	NOUN
ejpam-5048	202	10	,	,	PUNCT
ejpam-5048	202	11	1	1	NUM
ejpam-5048	202	12	)	)	PUNCT
ejpam-5048	202	13	which	which	PRON
ejpam-5048	202	14	is	be	AUX
ejpam-5048	202	15	given	give	VERB
ejpam-5048	202	16	in	in	ADP
ejpam-5048	202	17	example	example	NOUN
ejpam-5048	202	18	8	8	NUM
ejpam-5048	202	19	.	.	PUNCT
ejpam-5048	202	20	consider	consider	VERB
ejpam-5048	202	21	a	a	DET
ejpam-5048	202	22	subset	subset	NOUN
ejpam-5048	203	1	k	k	X
ejpam-5048	203	2	:	:	PUNCT
ejpam-5048	203	3	=	=	PUNCT
ejpam-5048	203	4	x×n0	x×n0	PROPN
ejpam-5048	203	5	of	of	ADP
ejpam-5048	203	6	y	y	PROPN
ejpam-5048	203	7	where	where	SCONJ
ejpam-5048	203	8	n0	n0	X
ejpam-5048	203	9	=	=	SYM
ejpam-5048	203	10	n∪{0	n∪{0	PROPN
ejpam-5048	203	11	}	}	PUNCT
ejpam-5048	203	12	and	and	CCONJ
ejpam-5048	203	13	n	n	PRON
ejpam-5048	203	14	is	be	AUX
ejpam-5048	203	15	the	the	DET
ejpam-5048	203	16	set	set	NOUN
ejpam-5048	203	17	of	of	ADP
ejpam-5048	203	18	all	all	DET
ejpam-5048	203	19	natural	natural	ADJ
ejpam-5048	203	20	numbers	number	NOUN
ejpam-5048	203	21	.	.	PUNCT
ejpam-5048	204	1	it	it	PRON
ejpam-5048	204	2	is	be	AUX
ejpam-5048	204	3	clear	clear	ADJ
ejpam-5048	204	4	that	that	SCONJ
ejpam-5048	204	5	1	1	X
ejpam-5048	204	6	=	=	SYM
ejpam-5048	204	7	(	(	PUNCT
ejpam-5048	204	8	1x	1x	NUM
ejpam-5048	204	9	,	,	PUNCT
ejpam-5048	204	10	0	0	NUM
ejpam-5048	204	11	)	)	PUNCT
ejpam-5048	204	12	∈	∈	PROPN
ejpam-5048	204	13	k.	k.	PROPN
ejpam-5048	204	14	let	let	VERB
ejpam-5048	204	15	(	(	PUNCT
ejpam-5048	204	16	κ1	κ1	NOUN
ejpam-5048	204	17	,	,	PUNCT
ejpam-5048	204	18	ϖ1	ϖ1	PROPN
ejpam-5048	204	19	)	)	PUNCT
ejpam-5048	204	20	,	,	PUNCT
ejpam-5048	204	21	(	(	PUNCT
ejpam-5048	204	22	κ2	κ2	PROPN
ejpam-5048	204	23	,	,	PUNCT
ejpam-5048	204	24	ϖ2	ϖ2	NOUN
ejpam-5048	204	25	)	)	PUNCT
ejpam-5048	204	26	∈	∈	PROPN
ejpam-5048	205	1	y	y	NOUN
ejpam-5048	205	2	be	be	AUX
ejpam-5048	205	3	such	such	ADJ
ejpam-5048	205	4	that	that	SCONJ
ejpam-5048	205	5	(	(	PUNCT
ejpam-5048	205	6	κ1	κ1	NOUN
ejpam-5048	205	7	,	,	PUNCT
ejpam-5048	205	8	ϖ1	ϖ1	PROPN
ejpam-5048	205	9	)	)	PUNCT
ejpam-5048	205	10	∗	∗	NOUN
ejpam-5048	205	11	(	(	PUNCT
ejpam-5048	205	12	κ2	κ2	PROPN
ejpam-5048	205	13	,	,	PUNCT
ejpam-5048	205	14	ϖ2	ϖ2	NOUN
ejpam-5048	205	15	)	)	PUNCT
ejpam-5048	205	16	∈	∈	PROPN
ejpam-5048	206	1	k	k	PROPN
ejpam-5048	206	2	and	and	CCONJ
ejpam-5048	206	3	(	(	PUNCT
ejpam-5048	206	4	κ1	κ1	NOUN
ejpam-5048	206	5	,	,	PUNCT
ejpam-5048	206	6	ϖ1	ϖ1	ADJ
ejpam-5048	206	7	)	)	PUNCT
ejpam-5048	206	8	∈	∈	PROPN
ejpam-5048	207	1	k.	k.	PROPN
ejpam-5048	208	1	then	then	ADV
ejpam-5048	208	2	(	(	PUNCT
ejpam-5048	208	3	κ1	κ1	NOUN
ejpam-5048	208	4	,	,	PUNCT
ejpam-5048	208	5	ϖ1	ϖ1	PROPN
ejpam-5048	208	6	)	)	PUNCT
ejpam-5048	208	7	∗	∗	NOUN
ejpam-5048	208	8	(	(	PUNCT
ejpam-5048	208	9	κ2	κ2	PROPN
ejpam-5048	208	10	,	,	PUNCT
ejpam-5048	208	11	ϖ2	ϖ2	NOUN
ejpam-5048	208	12	)	)	PUNCT
ejpam-5048	208	13	=	=	PUNCT
ejpam-5048	208	14	(	(	PUNCT
ejpam-5048	208	15	κ1	κ1	NOUN
ejpam-5048	208	16	∗x	∗x	PROPN
ejpam-5048	208	17	κ2	κ2	NOUN
ejpam-5048	208	18	,	,	PUNCT
ejpam-5048	208	19	ϖ2	ϖ2	NOUN
ejpam-5048	208	20	−ϖ1	−ϖ1	NOUN
ejpam-5048	208	21	)	)	PUNCT
ejpam-5048	208	22	∈	∈	PROPN
ejpam-5048	209	1	k	k	NOUN
ejpam-5048	209	2	,	,	PUNCT
ejpam-5048	209	3	and	and	CCONJ
ejpam-5048	209	4	so	so	ADV
ejpam-5048	209	5	ϖ1	ϖ1	PROPN
ejpam-5048	209	6	∈	∈	PROPN
ejpam-5048	209	7	n0	n0	NOUN
ejpam-5048	209	8	and	and	CCONJ
ejpam-5048	209	9	ϖ2	ϖ2	NOUN
ejpam-5048	209	10	−ϖ1	−ϖ1	PROPN
ejpam-5048	209	11	∈	∈	PROPN
ejpam-5048	209	12	n0	n0	PROPN
ejpam-5048	209	13	.	.	PUNCT
ejpam-5048	210	1	hence	hence	ADV
ejpam-5048	210	2	ϖ2	ϖ2	PROPN
ejpam-5048	210	3	∈	∈	PROPN
ejpam-5048	210	4	n0	n0	PROPN
ejpam-5048	210	5	,	,	PUNCT
ejpam-5048	210	6	and	and	CCONJ
ejpam-5048	210	7	thus	thus	ADV
ejpam-5048	210	8	(	(	PUNCT
ejpam-5048	210	9	κ2	κ2	NOUN
ejpam-5048	210	10	,	,	PUNCT
ejpam-5048	210	11	ϖ2	ϖ2	NOUN
ejpam-5048	210	12	)	)	PUNCT
ejpam-5048	210	13	∈	∈	PROPN
ejpam-5048	211	1	k.	k.	PROPN
ejpam-5048	211	2	therefore	therefore	ADV
ejpam-5048	211	3	k	k	PROPN
ejpam-5048	211	4	is	be	AUX
ejpam-5048	211	5	a	a	DET
ejpam-5048	211	6	qge	qge	NOUN
ejpam-5048	211	7	-	-	NOUN
ejpam-5048	211	8	filter	filter	NOUN
ejpam-5048	211	9	of	of	ADP
ejpam-5048	211	10	y	y	PROPN
ejpam-5048	211	11	.	.	PUNCT
ejpam-5048	212	1	theorem	theorem	ADJ
ejpam-5048	212	2	5	5	NUM
ejpam-5048	212	3	.	.	PUNCT
ejpam-5048	213	1	every	every	DET
ejpam-5048	213	2	qge	qge	NOUN
ejpam-5048	213	3	-	-	NOUN
ejpam-5048	213	4	filter	filter	NOUN
ejpam-5048	213	5	f	f	NOUN
ejpam-5048	213	6	of	of	ADP
ejpam-5048	213	7	x	x	PUNCT
ejpam-5048	213	8	satisfies	satisfie	NOUN
ejpam-5048	213	9	:	:	PUNCT
ejpam-5048	213	10	(	(	PUNCT
ejpam-5048	213	11	∀ϖ,π	∀ϖ,π	SYM
ejpam-5048	213	12	∈	∈	PROPN
ejpam-5048	213	13	f	f	PROPN
ejpam-5048	213	14	)	)	PUNCT
ejpam-5048	213	15	(	(	PUNCT
ejpam-5048	213	16	q(ϖ,π	q(ϖ,π	NUM
ejpam-5048	213	17	)	)	PUNCT
ejpam-5048	213	18	:	:	PUNCT
ejpam-5048	213	19	=	=	X
ejpam-5048	213	20	{	{	PUNCT
ejpam-5048	213	21	κ	κ	X
ejpam-5048	213	22	∈	∈	PROPN
ejpam-5048	213	23	x	x	INTJ
ejpam-5048	214	1	|	|	ADV
ejpam-5048	214	2	ϖ	ϖ	INTJ
ejpam-5048	214	3	∗	∗	NOUN
ejpam-5048	214	4	κ	κ	NOUN
ejpam-5048	214	5	=	=	SYM
ejpam-5048	214	6	π	π	PROPN
ejpam-5048	214	7	}	}	PUNCT
ejpam-5048	214	8	⊆	⊆	NUM
ejpam-5048	214	9	f	f	NOUN
ejpam-5048	214	10	)	)	PUNCT
ejpam-5048	214	11	,	,	PUNCT
ejpam-5048	214	12	(	(	PUNCT
ejpam-5048	214	13	24	24	NUM
ejpam-5048	214	14	)	)	PUNCT
ejpam-5048	214	15	proof	proof	NOUN
ejpam-5048	214	16	.	.	PUNCT
ejpam-5048	215	1	assume	assume	VERB
ejpam-5048	215	2	that	that	SCONJ
ejpam-5048	215	3	f	f	PROPN
ejpam-5048	215	4	is	be	AUX
ejpam-5048	215	5	a	a	DET
ejpam-5048	215	6	qge	qge	NOUN
ejpam-5048	215	7	-	-	NOUN
ejpam-5048	215	8	filter	filter	NOUN
ejpam-5048	215	9	of	of	ADP
ejpam-5048	215	10	x	x	PUNCT
ejpam-5048	215	11	and	and	CCONJ
ejpam-5048	215	12	let	let	VERB
ejpam-5048	215	13	κ	κ	PROPN
ejpam-5048	215	14	∈	∈	PROPN
ejpam-5048	215	15	q(ϖ,π	q(ϖ,π	NUM
ejpam-5048	215	16	)	)	PUNCT
ejpam-5048	215	17	for	for	ADP
ejpam-5048	215	18	ϖ,π	ϖ,π	PROPN
ejpam-5048	215	19	∈	∈	PROPN
ejpam-5048	215	20	f	f	X
ejpam-5048	215	21	.	.	PUNCT
ejpam-5048	216	1	then	then	ADV
ejpam-5048	216	2	ϖ	ϖ	PRON
ejpam-5048	216	3	∗	∗	NOUN
ejpam-5048	216	4	κ	κ	X
ejpam-5048	217	1	=	=	SYM
ejpam-5048	217	2	π	π	X
ejpam-5048	217	3	∈	∈	PROPN
ejpam-5048	217	4	f	f	PROPN
ejpam-5048	217	5	and	and	CCONJ
ejpam-5048	217	6	so	so	ADV
ejpam-5048	217	7	κ	κ	PROPN
ejpam-5048	217	8	∈	∈	PROPN
ejpam-5048	217	9	f	f	X
ejpam-5048	217	10	.	.	PUNCT
ejpam-5048	218	1	hence	hence	ADV
ejpam-5048	218	2	q(ϖ,π	q(ϖ,π	NUM
ejpam-5048	218	3	)	)	PUNCT
ejpam-5048	218	4	⊆	⊆	NUM
ejpam-5048	218	5	f	f	NOUN
ejpam-5048	218	6	.	.	PUNCT
ejpam-5048	219	1	proposition	proposition	NOUN
ejpam-5048	219	2	4	4	NUM
ejpam-5048	219	3	.	.	PUNCT
ejpam-5048	220	1	if	if	SCONJ
ejpam-5048	220	2	f	f	PROPN
ejpam-5048	220	3	is	be	AUX
ejpam-5048	220	4	a	a	DET
ejpam-5048	220	5	subset	subset	NOUN
ejpam-5048	220	6	of	of	ADP
ejpam-5048	220	7	x	x	PRON
ejpam-5048	220	8	that	that	PRON
ejpam-5048	220	9	satisfies	satisfy	VERB
ejpam-5048	220	10	the	the	DET
ejpam-5048	220	11	condition	condition	NOUN
ejpam-5048	220	12	(	(	PUNCT
ejpam-5048	220	13	24	24	NUM
ejpam-5048	220	14	)	)	PUNCT
ejpam-5048	220	15	,	,	PUNCT
ejpam-5048	220	16	then	then	ADV
ejpam-5048	220	17	f	f	PROPN
ejpam-5048	220	18	satisfies	satisfy	VERB
ejpam-5048	220	19	the	the	DET
ejpam-5048	220	20	condition	condition	NOUN
ejpam-5048	220	21	(	(	PUNCT
ejpam-5048	220	22	23	23	NUM
ejpam-5048	220	23	)	)	PUNCT
ejpam-5048	220	24	.	.	PUNCT
ejpam-5048	221	1	proof	proof	NOUN
ejpam-5048	221	2	.	.	PUNCT
ejpam-5048	222	1	let	let	VERB
ejpam-5048	222	2	f	f	PRON
ejpam-5048	222	3	be	be	AUX
ejpam-5048	222	4	a	a	DET
ejpam-5048	222	5	subset	subset	NOUN
ejpam-5048	222	6	of	of	ADP
ejpam-5048	222	7	x	x	PRON
ejpam-5048	222	8	that	that	PRON
ejpam-5048	222	9	satisfies	satisfy	VERB
ejpam-5048	222	10	the	the	DET
ejpam-5048	222	11	condition	condition	NOUN
ejpam-5048	222	12	(	(	PUNCT
ejpam-5048	222	13	24	24	NUM
ejpam-5048	222	14	)	)	PUNCT
ejpam-5048	222	15	.	.	PUNCT
ejpam-5048	223	1	let	let	VERB
ejpam-5048	223	2	κ	κ	VERB
ejpam-5048	223	3	,	,	PUNCT
ejpam-5048	223	4	δ	δ	PROPN
ejpam-5048	223	5	∈	∈	PROPN
ejpam-5048	223	6	x	x	AUX
ejpam-5048	223	7	be	be	AUX
ejpam-5048	223	8	such	such	ADJ
ejpam-5048	223	9	that	that	SCONJ
ejpam-5048	223	10	κ	κ	PROPN
ejpam-5048	223	11	∈	∈	PROPN
ejpam-5048	223	12	f	f	PROPN
ejpam-5048	223	13	and	and	CCONJ
ejpam-5048	223	14	κ	κ	PROPN
ejpam-5048	223	15	∗	∗	NOUN
ejpam-5048	223	16	δ	δ	PROPN
ejpam-5048	224	1	∈	∈	PROPN
ejpam-5048	224	2	f	f	PROPN
ejpam-5048	224	3	.	.	PUNCT
ejpam-5048	225	1	then	then	ADV
ejpam-5048	225	2	the	the	DET
ejpam-5048	225	3	equality	equality	NOUN
ejpam-5048	225	4	κ	κ	ADP
ejpam-5048	225	5	∗	∗	NOUN
ejpam-5048	225	6	δ	δ	X
ejpam-5048	225	7	=	=	SYM
ejpam-5048	225	8	κ	κ	PROPN
ejpam-5048	225	9	∗	∗	NOUN
ejpam-5048	225	10	δ	δ	PROPN
ejpam-5048	225	11	induces	induce	VERB
ejpam-5048	225	12	δ	δ	PROPN
ejpam-5048	225	13	∈	∈	PROPN
ejpam-5048	225	14	q(κ	q(κ	PROPN
ejpam-5048	225	15	,	,	PUNCT
ejpam-5048	225	16	κ	κ	PROPN
ejpam-5048	225	17	∗	∗	X
ejpam-5048	225	18	δ	δ	PROPN
ejpam-5048	225	19	)	)	PUNCT
ejpam-5048	225	20	⊆	⊆	NUM
ejpam-5048	225	21	f	f	NOUN
ejpam-5048	225	22	,	,	PUNCT
ejpam-5048	225	23	and	and	CCONJ
ejpam-5048	225	24	so	so	ADV
ejpam-5048	225	25	f	f	PROPN
ejpam-5048	225	26	satisfies	satisfy	VERB
ejpam-5048	225	27	the	the	DET
ejpam-5048	225	28	condition	condition	NOUN
ejpam-5048	225	29	(	(	PUNCT
ejpam-5048	225	30	23	23	NUM
ejpam-5048	225	31	)	)	PUNCT
ejpam-5048	225	32	.	.	PUNCT
ejpam-5048	226	1	we	we	PRON
ejpam-5048	226	2	present	present	VERB
ejpam-5048	226	3	the	the	DET
ejpam-5048	226	4	following	follow	VERB
ejpam-5048	226	5	open	open	ADJ
ejpam-5048	226	6	question	question	NOUN
ejpam-5048	226	7	.	.	PUNCT
ejpam-5048	227	1	question	question	NOUN
ejpam-5048	227	2	4	4	NUM
ejpam-5048	227	3	.	.	PUNCT
ejpam-5048	228	1	if	if	SCONJ
ejpam-5048	228	2	f	f	PROPN
ejpam-5048	228	3	is	be	AUX
ejpam-5048	228	4	a	a	DET
ejpam-5048	228	5	subset	subset	NOUN
ejpam-5048	228	6	of	of	ADP
ejpam-5048	228	7	x	x	PRON
ejpam-5048	228	8	that	that	PRON
ejpam-5048	228	9	satisfies	satisfy	VERB
ejpam-5048	228	10	the	the	DET
ejpam-5048	228	11	condition	condition	NOUN
ejpam-5048	228	12	(	(	PUNCT
ejpam-5048	228	13	24	24	NUM
ejpam-5048	228	14	)	)	PUNCT
ejpam-5048	228	15	,	,	PUNCT
ejpam-5048	228	16	then	then	ADV
ejpam-5048	228	17	does	do	AUX
ejpam-5048	228	18	f	f	PROPN
ejpam-5048	228	19	include	include	VERB
ejpam-5048	228	20	the	the	DET
ejpam-5048	228	21	unit	unit	NOUN
ejpam-5048	228	22	1	1	NUM
ejpam-5048	228	23	?	?	PUNCT
ejpam-5048	229	1	if	if	SCONJ
ejpam-5048	229	2	we	we	PRON
ejpam-5048	229	3	can	can	AUX
ejpam-5048	229	4	get	get	VERB
ejpam-5048	229	5	the	the	DET
ejpam-5048	229	6	positive	positive	ADJ
ejpam-5048	229	7	answer	answer	NOUN
ejpam-5048	229	8	to	to	ADP
ejpam-5048	229	9	the	the	DET
ejpam-5048	229	10	question	question	NOUN
ejpam-5048	229	11	4	4	NUM
ejpam-5048	229	12	,	,	PUNCT
ejpam-5048	229	13	then	then	ADV
ejpam-5048	229	14	we	we	PRON
ejpam-5048	229	15	know	know	VERB
ejpam-5048	229	16	that	that	SCONJ
ejpam-5048	229	17	every	every	DET
ejpam-5048	229	18	subset	subset	NOUN
ejpam-5048	229	19	f	f	PROPN
ejpam-5048	229	20	of	of	ADP
ejpam-5048	229	21	x	x	PRON
ejpam-5048	229	22	which	which	PRON
ejpam-5048	229	23	satisfies	satisfy	VERB
ejpam-5048	229	24	the	the	DET
ejpam-5048	229	25	condition	condition	NOUN
ejpam-5048	229	26	(	(	PUNCT
ejpam-5048	229	27	24	24	NUM
ejpam-5048	229	28	)	)	PUNCT
ejpam-5048	229	29	is	be	AUX
ejpam-5048	229	30	a	a	DET
ejpam-5048	229	31	qge	qge	NOUN
ejpam-5048	229	32	-	-	NOUN
ejpam-5048	229	33	filter	filter	NOUN
ejpam-5048	229	34	of	of	ADP
ejpam-5048	229	35	x.	x.	NOUN
ejpam-5048	229	36	if	if	SCONJ
ejpam-5048	229	37	f	f	PROPN
ejpam-5048	229	38	is	be	AUX
ejpam-5048	229	39	a	a	DET
ejpam-5048	229	40	subset	subset	NOUN
ejpam-5048	229	41	of	of	ADP
ejpam-5048	229	42	x	x	PRON
ejpam-5048	229	43	that	that	PRON
ejpam-5048	229	44	satisfies	satisfy	VERB
ejpam-5048	229	45	the	the	DET
ejpam-5048	229	46	condition	condition	NOUN
ejpam-5048	229	47	(	(	PUNCT
ejpam-5048	229	48	24	24	NUM
ejpam-5048	229	49	)	)	PUNCT
ejpam-5048	229	50	for	for	ADP
ejpam-5048	229	51	all	all	DET
ejpam-5048	229	52	ϖ,π	ϖ,π	ADJ
ejpam-5048	229	53	∈	∈	PROPN
ejpam-5048	229	54	x	x	PUNCT
ejpam-5048	229	55	with	with	ADP
ejpam-5048	229	56	ϖ	ϖ	X
ejpam-5048	229	57	̸=	̸=	PROPN
ejpam-5048	229	58	π	π	PROPN
ejpam-5048	229	59	,	,	PUNCT
ejpam-5048	229	60	then	then	ADV
ejpam-5048	229	61	f	f	PROPN
ejpam-5048	229	62	may	may	AUX
ejpam-5048	229	63	not	not	PART
ejpam-5048	229	64	be	be	AUX
ejpam-5048	229	65	a	a	DET
ejpam-5048	229	66	qge	qge	NOUN
ejpam-5048	229	67	-	-	NOUN
ejpam-5048	229	68	filter	filter	NOUN
ejpam-5048	229	69	of	of	ADP
ejpam-5048	229	70	x	x	PART
ejpam-5048	229	71	as	as	SCONJ
ejpam-5048	229	72	shown	show	VERB
ejpam-5048	229	73	in	in	ADP
ejpam-5048	229	74	the	the	DET
ejpam-5048	229	75	following	follow	VERB
ejpam-5048	229	76	example	example	NOUN
ejpam-5048	229	77	.	.	PUNCT
ejpam-5048	230	1	y.	y.	PROPN
ejpam-5048	230	2	b.	b.	PROPN
ejpam-5048	230	3	jun	jun	PROPN
ejpam-5048	230	4	,	,	PUNCT
ejpam-5048	230	5	ravikumar	ravikumar	PROPN
ejpam-5048	230	6	bandaru	bandaru	PROPN
ejpam-5048	230	7	,	,	PUNCT
ejpam-5048	230	8	rahul	rahul	PROPN
ejpam-5048	230	9	shukla	shukla	PROPN
ejpam-5048	230	10	/	/	SYM
ejpam-5048	230	11	eur	eur	PROPN
ejpam-5048	230	12	.	.	PUNCT
ejpam-5048	231	1	j.	j.	PROPN
ejpam-5048	231	2	pure	pure	PROPN
ejpam-5048	231	3	appl	appl	PROPN
ejpam-5048	231	4	.	.	PROPN
ejpam-5048	231	5	math	math	PROPN
ejpam-5048	231	6	,	,	PUNCT
ejpam-5048	231	7	17	17	NUM
ejpam-5048	231	8	(	(	PUNCT
ejpam-5048	231	9	1	1	NUM
ejpam-5048	231	10	)	)	PUNCT
ejpam-5048	231	11	(	(	PUNCT
ejpam-5048	231	12	2024	2024	NUM
ejpam-5048	231	13	)	)	PUNCT
ejpam-5048	231	14	,	,	PUNCT
ejpam-5048	231	15	569	569	NUM
ejpam-5048	231	16	-	-	SYM
ejpam-5048	231	17	581	581	NUM
ejpam-5048	231	18	577	577	NUM
ejpam-5048	231	19	example	example	NOUN
ejpam-5048	231	20	15	15	NUM
ejpam-5048	231	21	.	.	PUNCT
ejpam-5048	231	22	consider	consider	VERB
ejpam-5048	231	23	the	the	DET
ejpam-5048	231	24	qge	qge	NOUN
ejpam-5048	231	25	-	-	NOUN
ejpam-5048	231	26	algebra	algebra	NOUN
ejpam-5048	231	27	(	(	PUNCT
ejpam-5048	231	28	x	x	X
ejpam-5048	231	29	,	,	PUNCT
ejpam-5048	231	30	∗	∗	NOUN
ejpam-5048	231	31	,	,	PUNCT
ejpam-5048	231	32	1	1	NUM
ejpam-5048	231	33	)	)	PUNCT
ejpam-5048	231	34	given	give	VERB
ejpam-5048	231	35	in	in	ADP
ejpam-5048	231	36	example	example	NOUN
ejpam-5048	231	37	2	2	NUM
ejpam-5048	231	38	.	.	PUNCT
ejpam-5048	232	1	let	let	VERB
ejpam-5048	232	2	f	f	PROPN
ejpam-5048	232	3	=	=	PUNCT
ejpam-5048	232	4	{	{	PUNCT
ejpam-5048	232	5	1	1	NUM
ejpam-5048	232	6	,	,	PUNCT
ejpam-5048	232	7	d	d	NOUN
ejpam-5048	232	8	}	}	PUNCT
ejpam-5048	232	9	.	.	PUNCT
ejpam-5048	233	1	then	then	ADV
ejpam-5048	233	2	we	we	PRON
ejpam-5048	233	3	can	can	AUX
ejpam-5048	233	4	observe	observe	VERB
ejpam-5048	233	5	that	that	SCONJ
ejpam-5048	233	6	q(1	q(1	NOUN
ejpam-5048	233	7	,	,	PUNCT
ejpam-5048	233	8	d	d	NOUN
ejpam-5048	233	9	)	)	PUNCT
ejpam-5048	233	10	=	=	SYM
ejpam-5048	233	11	q(d	q(d	PROPN
ejpam-5048	233	12	,	,	PUNCT
ejpam-5048	233	13	1	1	NUM
ejpam-5048	233	14	)	)	PUNCT
ejpam-5048	233	15	=	=	PRON
ejpam-5048	233	16	{	{	PUNCT
ejpam-5048	233	17	d	d	NOUN
ejpam-5048	233	18	}	}	PUNCT
ejpam-5048	233	19	⊆	⊆	NUM
ejpam-5048	233	20	f	f	NOUN
ejpam-5048	233	21	for	for	ADP
ejpam-5048	233	22	all	all	DET
ejpam-5048	233	23	1	1	NUM
ejpam-5048	233	24	,	,	PUNCT
ejpam-5048	233	25	d	d	PROPN
ejpam-5048	233	26	∈	∈	PROPN
ejpam-5048	233	27	f	f	X
ejpam-5048	233	28	.	.	PUNCT
ejpam-5048	234	1	but	but	CCONJ
ejpam-5048	234	2	f	f	PROPN
ejpam-5048	234	3	is	be	AUX
ejpam-5048	234	4	not	not	PART
ejpam-5048	234	5	a	a	DET
ejpam-5048	234	6	qge	qge	NOUN
ejpam-5048	234	7	-	-	NOUN
ejpam-5048	234	8	filter	filter	NOUN
ejpam-5048	234	9	of	of	ADP
ejpam-5048	234	10	x	x	SYM
ejpam-5048	234	11	since	since	SCONJ
ejpam-5048	234	12	d	d	PROPN
ejpam-5048	234	13	∈	∈	PROPN
ejpam-5048	234	14	f	f	PROPN
ejpam-5048	234	15	and	and	CCONJ
ejpam-5048	234	16	d	d	PROPN
ejpam-5048	234	17	∗	∗	NOUN
ejpam-5048	234	18	b	b	X
ejpam-5048	235	1	=	=	SYM
ejpam-5048	235	2	d	d	PROPN
ejpam-5048	235	3	∈	∈	PROPN
ejpam-5048	235	4	f	f	PROPN
ejpam-5048	235	5	but	but	CCONJ
ejpam-5048	235	6	b	b	PROPN
ejpam-5048	235	7	/∈	/∈	PROPN
ejpam-5048	236	1	f	f	PROPN
ejpam-5048	236	2	.	.	PUNCT
ejpam-5048	237	1	question	question	NOUN
ejpam-5048	237	2	5	5	NUM
ejpam-5048	237	3	.	.	PUNCT
ejpam-5048	238	1	does	do	AUX
ejpam-5048	238	2	any	any	DET
ejpam-5048	238	3	qge	qge	NOUN
ejpam-5048	238	4	-	-	NOUN
ejpam-5048	238	5	filter	filter	ADJ
ejpam-5048	238	6	f	f	NOUN
ejpam-5048	238	7	of	of	ADP
ejpam-5048	238	8	x	x	PUNCT
ejpam-5048	238	9	satisfy	satisfy	VERB
ejpam-5048	238	10	the	the	DET
ejpam-5048	238	11	condition	condition	NOUN
ejpam-5048	238	12	below	below	ADV
ejpam-5048	238	13	?	?	PUNCT
ejpam-5048	239	1	(	(	PUNCT
ejpam-5048	239	2	∀κ	∀κ	NUM
ejpam-5048	239	3	,	,	PUNCT
ejpam-5048	239	4	δ	δ	PROPN
ejpam-5048	239	5	,	,	PUNCT
ejpam-5048	239	6	ς	ς	PROPN
ejpam-5048	239	7	∈	∈	PROPN
ejpam-5048	239	8	x)(ς	x)(ς	PUNCT
ejpam-5048	239	9	∗	∗	NOUN
ejpam-5048	239	10	(	(	PUNCT
ejpam-5048	239	11	δ	δ	PROPN
ejpam-5048	239	12	∗	∗	PROPN
ejpam-5048	239	13	κ	κ	NOUN
ejpam-5048	239	14	)	)	PUNCT
ejpam-5048	239	15	∈	∈	PROPN
ejpam-5048	240	1	f	f	X
ejpam-5048	240	2	,	,	PUNCT
ejpam-5048	240	3	ς	ς	PROPN
ejpam-5048	240	4	∗	∗	NOUN
ejpam-5048	240	5	δ	δ	NOUN
ejpam-5048	240	6	∈	∈	PROPN
ejpam-5048	240	7	f	f	PROPN
ejpam-5048	240	8	⇒	⇒	PROPN
ejpam-5048	240	9	ς	ς	PROPN
ejpam-5048	240	10	∗	∗	NOUN
ejpam-5048	240	11	κ	κ	ADP
ejpam-5048	240	12	∈	∈	PROPN
ejpam-5048	240	13	f	f	PROPN
ejpam-5048	240	14	)	)	PUNCT
ejpam-5048	240	15	.	.	PUNCT
ejpam-5048	241	1	(	(	PUNCT
ejpam-5048	241	2	25	25	NUM
ejpam-5048	241	3	)	)	PUNCT
ejpam-5048	241	4	the	the	DET
ejpam-5048	241	5	example	example	NOUN
ejpam-5048	241	6	below	below	ADP
ejpam-5048	241	7	shows	show	VERB
ejpam-5048	241	8	that	that	SCONJ
ejpam-5048	241	9	the	the	DET
ejpam-5048	241	10	answer	answer	NOUN
ejpam-5048	241	11	to	to	PART
ejpam-5048	241	12	question	question	NOUN
ejpam-5048	241	13	5	5	NUM
ejpam-5048	241	14	is	be	AUX
ejpam-5048	241	15	negative	negative	ADJ
ejpam-5048	241	16	.	.	PUNCT
ejpam-5048	242	1	example	example	NOUN
ejpam-5048	243	1	16	16	NUM
ejpam-5048	243	2	.	.	PUNCT
ejpam-5048	244	1	let	let	VERB
ejpam-5048	244	2	x	x	PUNCT
ejpam-5048	244	3	=	=	PRON
ejpam-5048	244	4	{	{	PUNCT
ejpam-5048	244	5	1	1	NUM
ejpam-5048	244	6	,	,	PUNCT
ejpam-5048	244	7	a	a	DET
ejpam-5048	244	8	,	,	PUNCT
ejpam-5048	244	9	b	b	NOUN
ejpam-5048	244	10	,	,	PUNCT
ejpam-5048	244	11	c	c	NOUN
ejpam-5048	244	12	,	,	PUNCT
ejpam-5048	244	13	d	d	NOUN
ejpam-5048	244	14	,	,	PUNCT
ejpam-5048	244	15	e	e	AUX
ejpam-5048	244	16	}	}	PUNCT
ejpam-5048	244	17	be	be	AUX
ejpam-5048	244	18	a	a	DET
ejpam-5048	244	19	set	set	NOUN
ejpam-5048	244	20	with	with	ADP
ejpam-5048	244	21	a	a	DET
ejpam-5048	244	22	binary	binary	ADJ
ejpam-5048	244	23	operation	operation	NOUN
ejpam-5048	244	24	“	"	PUNCT
ejpam-5048	244	25	∗	∗	NOUN
ejpam-5048	244	26	”	"	PUNCT
ejpam-5048	244	27	given	give	VERB
ejpam-5048	244	28	in	in	ADP
ejpam-5048	244	29	the	the	DET
ejpam-5048	244	30	following	follow	VERB
ejpam-5048	244	31	table	table	NOUN
ejpam-5048	244	32	:	:	PUNCT
ejpam-5048	244	33	∗	∗	NOUN
ejpam-5048	244	34	1	1	NUM
ejpam-5048	244	35	a	a	DET
ejpam-5048	244	36	b	b	NOUN
ejpam-5048	244	37	c	c	NOUN
ejpam-5048	244	38	d	d	X
ejpam-5048	244	39	e	e	PROPN
ejpam-5048	244	40	1	1	NUM
ejpam-5048	244	41	1	1	NUM
ejpam-5048	244	42	a	a	DET
ejpam-5048	244	43	b	b	NOUN
ejpam-5048	244	44	c	c	NOUN
ejpam-5048	245	1	d	d	PROPN
ejpam-5048	245	2	e	e	PROPN
ejpam-5048	245	3	a	a	DET
ejpam-5048	245	4	a	a	DET
ejpam-5048	245	5	1	1	NUM
ejpam-5048	245	6	d	d	NOUN
ejpam-5048	245	7	e	e	PROPN
ejpam-5048	245	8	b	b	PROPN
ejpam-5048	245	9	c	c	PROPN
ejpam-5048	245	10	b	b	PROPN
ejpam-5048	245	11	c	c	NOUN
ejpam-5048	245	12	d	d	SYM
ejpam-5048	245	13	1	1	PROPN
ejpam-5048	245	14	b	b	PROPN
ejpam-5048	245	15	e	e	X
ejpam-5048	245	16	a	a	PROPN
ejpam-5048	245	17	c	c	PROPN
ejpam-5048	245	18	b	b	PROPN
ejpam-5048	245	19	e	e	NOUN
ejpam-5048	245	20	c	c	PROPN
ejpam-5048	245	21	1	1	NUM
ejpam-5048	245	22	a	a	DET
ejpam-5048	245	23	d	d	X
ejpam-5048	245	24	d	d	PROPN
ejpam-5048	245	25	d	d	X
ejpam-5048	245	26	c	c	NOUN
ejpam-5048	245	27	e	e	NOUN
ejpam-5048	245	28	a	a	DET
ejpam-5048	245	29	1	1	NUM
ejpam-5048	245	30	b	b	NOUN
ejpam-5048	245	31	e	e	X
ejpam-5048	245	32	e	e	PROPN
ejpam-5048	245	33	b	b	PROPN
ejpam-5048	245	34	a	a	PRON
ejpam-5048	245	35	d	d	X
ejpam-5048	245	36	c	c	NOUN
ejpam-5048	245	37	1	1	NUM
ejpam-5048	245	38	then	then	ADV
ejpam-5048	245	39	(	(	PUNCT
ejpam-5048	245	40	x	x	X
ejpam-5048	245	41	,	,	PUNCT
ejpam-5048	245	42	∗	∗	NOUN
ejpam-5048	245	43	,	,	PUNCT
ejpam-5048	245	44	1	1	NUM
ejpam-5048	245	45	)	)	PUNCT
ejpam-5048	245	46	is	be	AUX
ejpam-5048	245	47	a	a	DET
ejpam-5048	245	48	qge	qge	NOUN
ejpam-5048	245	49	-	-	NOUN
ejpam-5048	245	50	algebra	algebra	NOUN
ejpam-5048	245	51	,	,	PUNCT
ejpam-5048	245	52	and	and	CCONJ
ejpam-5048	245	53	it	it	PRON
ejpam-5048	245	54	is	be	AUX
ejpam-5048	245	55	routine	routine	ADJ
ejpam-5048	245	56	to	to	PART
ejpam-5048	245	57	verify	verify	VERB
ejpam-5048	245	58	that	that	SCONJ
ejpam-5048	245	59	the	the	DET
ejpam-5048	245	60	set	set	NOUN
ejpam-5048	245	61	f	f	NOUN
ejpam-5048	245	62	:	:	PUNCT
ejpam-5048	245	63	=	=	SYM
ejpam-5048	245	64	{	{	PUNCT
ejpam-5048	245	65	1	1	NUM
ejpam-5048	245	66	,	,	PUNCT
ejpam-5048	245	67	b	b	NOUN
ejpam-5048	245	68	,	,	PUNCT
ejpam-5048	245	69	c	c	NOUN
ejpam-5048	245	70	}	}	PUNCT
ejpam-5048	245	71	is	be	AUX
ejpam-5048	245	72	a	a	DET
ejpam-5048	245	73	qge	qge	NOUN
ejpam-5048	245	74	-	-	NOUN
ejpam-5048	245	75	filter	filter	NOUN
ejpam-5048	245	76	of	of	ADP
ejpam-5048	245	77	x.	x.	NOUN
ejpam-5048	246	1	but	but	CCONJ
ejpam-5048	246	2	it	it	PRON
ejpam-5048	246	3	does	do	AUX
ejpam-5048	246	4	not	not	PART
ejpam-5048	246	5	satisfy	satisfy	VERB
ejpam-5048	246	6	(	(	PUNCT
ejpam-5048	246	7	25	25	NUM
ejpam-5048	246	8	)	)	PUNCT
ejpam-5048	246	9	since	since	SCONJ
ejpam-5048	246	10	a	a	DET
ejpam-5048	246	11	∗	∗	NOUN
ejpam-5048	246	12	(	(	PUNCT
ejpam-5048	246	13	a	a	DET
ejpam-5048	246	14	∗	∗	NOUN
ejpam-5048	246	15	1	1	NUM
ejpam-5048	246	16	)	)	PUNCT
ejpam-5048	246	17	=	=	PUNCT
ejpam-5048	246	18	a	a	DET
ejpam-5048	246	19	∗	∗	NOUN
ejpam-5048	246	20	a	a	DET
ejpam-5048	246	21	=	=	SYM
ejpam-5048	246	22	1	1	NUM
ejpam-5048	246	23	∈	∈	PROPN
ejpam-5048	246	24	f	f	NOUN
ejpam-5048	246	25	and	and	CCONJ
ejpam-5048	246	26	a	a	DET
ejpam-5048	246	27	∗	∗	NOUN
ejpam-5048	246	28	a	a	DET
ejpam-5048	246	29	=	=	SYM
ejpam-5048	246	30	1	1	NUM
ejpam-5048	246	31	∈	∈	PROPN
ejpam-5048	246	32	f	f	NOUN
ejpam-5048	246	33	,	,	PUNCT
ejpam-5048	246	34	but	but	CCONJ
ejpam-5048	246	35	a	a	DET
ejpam-5048	246	36	∗	∗	NOUN
ejpam-5048	246	37	1	1	NUM
ejpam-5048	246	38	=	=	SYM
ejpam-5048	246	39	a	a	PROPN
ejpam-5048	246	40	/∈	/∈	NOUN
ejpam-5048	246	41	f.	f.	NOUN
ejpam-5048	247	1	we	we	PRON
ejpam-5048	247	2	use	use	VERB
ejpam-5048	247	3	two	two	NUM
ejpam-5048	247	4	conditions	condition	NOUN
ejpam-5048	247	5	(	(	PUNCT
ejpam-5048	247	6	22	22	NUM
ejpam-5048	247	7	)	)	PUNCT
ejpam-5048	247	8	and	and	CCONJ
ejpam-5048	247	9	(	(	PUNCT
ejpam-5048	247	10	25	25	NUM
ejpam-5048	247	11	)	)	PUNCT
ejpam-5048	247	12	to	to	PART
ejpam-5048	247	13	make	make	VERB
ejpam-5048	247	14	a	a	DET
ejpam-5048	247	15	qge	qge	NOUN
ejpam-5048	247	16	-	-	NOUN
ejpam-5048	247	17	filter	filter	NOUN
ejpam-5048	247	18	from	from	ADP
ejpam-5048	247	19	a	a	DET
ejpam-5048	247	20	subset	subset	NOUN
ejpam-5048	247	21	.	.	PUNCT
ejpam-5048	248	1	theorem	theorem	NOUN
ejpam-5048	248	2	6	6	NUM
ejpam-5048	248	3	.	.	PUNCT
ejpam-5048	249	1	let	let	VERB
ejpam-5048	249	2	f	f	PRON
ejpam-5048	249	3	be	be	AUX
ejpam-5048	249	4	a	a	DET
ejpam-5048	249	5	subset	subset	NOUN
ejpam-5048	249	6	of	of	ADP
ejpam-5048	249	7	x	x	PRON
ejpam-5048	249	8	that	that	SCONJ
ejpam-5048	249	9	satisfies	satisfy	VERB
ejpam-5048	249	10	(	(	PUNCT
ejpam-5048	249	11	22	22	NUM
ejpam-5048	249	12	)	)	PUNCT
ejpam-5048	249	13	.	.	PUNCT
ejpam-5048	250	1	if	if	SCONJ
ejpam-5048	250	2	f	f	PROPN
ejpam-5048	250	3	satisfies	satisfy	VERB
ejpam-5048	250	4	the	the	DET
ejpam-5048	250	5	condition	condition	NOUN
ejpam-5048	250	6	(	(	PUNCT
ejpam-5048	250	7	25	25	NUM
ejpam-5048	250	8	)	)	PUNCT
ejpam-5048	250	9	,	,	PUNCT
ejpam-5048	250	10	then	then	ADV
ejpam-5048	250	11	it	it	PRON
ejpam-5048	250	12	is	be	AUX
ejpam-5048	250	13	a	a	DET
ejpam-5048	250	14	qge	qge	NOUN
ejpam-5048	250	15	-	-	NOUN
ejpam-5048	250	16	filter	filter	NOUN
ejpam-5048	250	17	of	of	ADP
ejpam-5048	250	18	x.	x.	NOUN
ejpam-5048	250	19	proof	proof	PROPN
ejpam-5048	250	20	.	.	PUNCT
ejpam-5048	251	1	assume	assume	VERB
ejpam-5048	251	2	that	that	SCONJ
ejpam-5048	251	3	a	a	DET
ejpam-5048	251	4	subset	subset	NOUN
ejpam-5048	251	5	f	f	X
ejpam-5048	251	6	of	of	ADP
ejpam-5048	251	7	x	x	PRON
ejpam-5048	251	8	satisfies	satisfy	VERB
ejpam-5048	251	9	two	two	NUM
ejpam-5048	251	10	conditions	condition	NOUN
ejpam-5048	251	11	(	(	PUNCT
ejpam-5048	251	12	22	22	NUM
ejpam-5048	251	13	)	)	PUNCT
ejpam-5048	251	14	and	and	CCONJ
ejpam-5048	251	15	(	(	PUNCT
ejpam-5048	251	16	25	25	NUM
ejpam-5048	251	17	)	)	PUNCT
ejpam-5048	251	18	.	.	PUNCT
ejpam-5048	252	1	let	let	VERB
ejpam-5048	252	2	κ	κ	VERB
ejpam-5048	252	3	,	,	PUNCT
ejpam-5048	252	4	δ	δ	PROPN
ejpam-5048	252	5	∈	∈	PROPN
ejpam-5048	252	6	x	x	AUX
ejpam-5048	252	7	be	be	AUX
ejpam-5048	252	8	such	such	ADJ
ejpam-5048	252	9	that	that	SCONJ
ejpam-5048	252	10	δ	δ	PROPN
ejpam-5048	252	11	∗	∗	VERB
ejpam-5048	252	12	κ	κ	X
ejpam-5048	252	13	∈	∈	PROPN
ejpam-5048	252	14	f	f	PROPN
ejpam-5048	252	15	and	and	CCONJ
ejpam-5048	252	16	δ	δ	PROPN
ejpam-5048	252	17	∈	∈	PROPN
ejpam-5048	253	1	f	f	X
ejpam-5048	253	2	.	.	PUNCT
ejpam-5048	254	1	if	if	SCONJ
ejpam-5048	254	2	we	we	PRON
ejpam-5048	254	3	take	take	VERB
ejpam-5048	254	4	ς	ς	PROPN
ejpam-5048	254	5	:	:	PUNCT
ejpam-5048	254	6	=	=	SYM
ejpam-5048	254	7	1	1	NUM
ejpam-5048	254	8	in	in	ADP
ejpam-5048	254	9	(	(	PUNCT
ejpam-5048	254	10	25	25	NUM
ejpam-5048	254	11	)	)	PUNCT
ejpam-5048	254	12	and	and	CCONJ
ejpam-5048	254	13	use	use	NOUN
ejpam-5048	254	14	(	(	PUNCT
ejpam-5048	254	15	ge2	ge2	NOUN
ejpam-5048	254	16	)	)	PUNCT
ejpam-5048	254	17	,	,	PUNCT
ejpam-5048	254	18	then	then	ADV
ejpam-5048	254	19	1	1	NUM
ejpam-5048	254	20	∗	∗	NOUN
ejpam-5048	254	21	(	(	PUNCT
ejpam-5048	254	22	δ	δ	PROPN
ejpam-5048	254	23	∗	∗	PROPN
ejpam-5048	254	24	κ	κ	NOUN
ejpam-5048	254	25	)	)	PUNCT
ejpam-5048	254	26	=	=	SYM
ejpam-5048	254	27	δ	δ	PROPN
ejpam-5048	254	28	∗	∗	VERB
ejpam-5048	254	29	κ	κ	X
ejpam-5048	254	30	∈	∈	PROPN
ejpam-5048	254	31	f	f	PROPN
ejpam-5048	254	32	and	and	CCONJ
ejpam-5048	254	33	1	1	NUM
ejpam-5048	254	34	∗	∗	NOUN
ejpam-5048	254	35	δ	δ	NOUN
ejpam-5048	254	36	=	=	PUNCT
ejpam-5048	254	37	δ	δ	PROPN
ejpam-5048	254	38	∈	∈	PROPN
ejpam-5048	254	39	f	f	X
ejpam-5048	254	40	.	.	PUNCT
ejpam-5048	255	1	it	it	PRON
ejpam-5048	255	2	follows	follow	VERB
ejpam-5048	255	3	from	from	ADP
ejpam-5048	255	4	(	(	PUNCT
ejpam-5048	255	5	25	25	NUM
ejpam-5048	255	6	)	)	PUNCT
ejpam-5048	255	7	and	and	CCONJ
ejpam-5048	255	8	use	use	NOUN
ejpam-5048	255	9	(	(	PUNCT
ejpam-5048	255	10	ge2	ge2	NOUN
ejpam-5048	255	11	)	)	PUNCT
ejpam-5048	255	12	that	that	PRON
ejpam-5048	255	13	κ	κ	AUX
ejpam-5048	255	14	=	=	SYM
ejpam-5048	255	15	1	1	NUM
ejpam-5048	255	16	∗	∗	NOUN
ejpam-5048	255	17	κ	κ	X
ejpam-5048	255	18	∈	∈	PROPN
ejpam-5048	255	19	f	f	X
ejpam-5048	255	20	.	.	PUNCT
ejpam-5048	256	1	thus	thus	ADV
ejpam-5048	256	2	f	f	PROPN
ejpam-5048	256	3	is	be	AUX
ejpam-5048	256	4	a	a	DET
ejpam-5048	256	5	qge	qge	NOUN
ejpam-5048	256	6	-	-	NOUN
ejpam-5048	256	7	filter	filter	NOUN
ejpam-5048	256	8	of	of	ADP
ejpam-5048	256	9	x.	x.	NOUN
ejpam-5048	256	10	for	for	ADP
ejpam-5048	256	11	a	a	DET
ejpam-5048	256	12	subset	subset	NOUN
ejpam-5048	256	13	f	f	PROPN
ejpam-5048	256	14	of	of	ADP
ejpam-5048	256	15	x	x	PRON
ejpam-5048	256	16	,	,	PUNCT
ejpam-5048	256	17	consider	consider	VERB
ejpam-5048	256	18	the	the	DET
ejpam-5048	256	19	condition	condition	NOUN
ejpam-5048	256	20	below	below	ADV
ejpam-5048	256	21	.	.	PUNCT
ejpam-5048	257	1	(	(	PUNCT
ejpam-5048	257	2	∀κ	∀κ	NUM
ejpam-5048	257	3	,	,	PUNCT
ejpam-5048	257	4	δ	δ	PROPN
ejpam-5048	257	5	,	,	PUNCT
ejpam-5048	257	6	ς	ς	PROPN
ejpam-5048	257	7	∈	∈	PROPN
ejpam-5048	257	8	x)(κ	x)(κ	PROPN
ejpam-5048	257	9	∗	∗	NOUN
ejpam-5048	257	10	(	(	PUNCT
ejpam-5048	257	11	δ	δ	PROPN
ejpam-5048	257	12	∗	∗	PROPN
ejpam-5048	257	13	ς	ς	NOUN
ejpam-5048	257	14	)	)	PUNCT
ejpam-5048	257	15	∈	∈	PROPN
ejpam-5048	257	16	f	f	PROPN
ejpam-5048	257	17	⇒	⇒	PROPN
ejpam-5048	257	18	δ	δ	PROPN
ejpam-5048	257	19	∗	∗	VERB
ejpam-5048	257	20	ς	ς	PROPN
ejpam-5048	257	21	∈	∈	PROPN
ejpam-5048	257	22	f	f	PROPN
ejpam-5048	257	23	)	)	PUNCT
ejpam-5048	257	24	.	.	PUNCT
ejpam-5048	258	1	(	(	PUNCT
ejpam-5048	258	2	26	26	NUM
ejpam-5048	258	3	)	)	PUNCT
ejpam-5048	258	4	the	the	DET
ejpam-5048	258	5	following	follow	VERB
ejpam-5048	258	6	example	example	NOUN
ejpam-5048	258	7	shows	show	VERB
ejpam-5048	258	8	that	that	SCONJ
ejpam-5048	258	9	a	a	DET
ejpam-5048	258	10	qge	qge	NOUN
ejpam-5048	258	11	-	-	NOUN
ejpam-5048	258	12	filter	filter	NOUN
ejpam-5048	258	13	f	f	NOUN
ejpam-5048	258	14	of	of	ADP
ejpam-5048	258	15	x	x	PRON
ejpam-5048	258	16	may	may	AUX
ejpam-5048	258	17	not	not	PART
ejpam-5048	258	18	satisfy	satisfy	VERB
ejpam-5048	258	19	the	the	DET
ejpam-5048	258	20	condition	condition	NOUN
ejpam-5048	258	21	(	(	PUNCT
ejpam-5048	258	22	26	26	NUM
ejpam-5048	258	23	)	)	PUNCT
ejpam-5048	258	24	.	.	PUNCT
ejpam-5048	259	1	example	example	NOUN
ejpam-5048	260	1	17	17	NUM
ejpam-5048	260	2	.	.	PUNCT
ejpam-5048	261	1	let	let	VERB
ejpam-5048	261	2	(	(	PUNCT
ejpam-5048	261	3	x	x	X
ejpam-5048	261	4	,	,	PUNCT
ejpam-5048	261	5	∗	∗	NOUN
ejpam-5048	261	6	,	,	PUNCT
ejpam-5048	261	7	1	1	NUM
ejpam-5048	261	8	)	)	PUNCT
ejpam-5048	261	9	be	be	AUX
ejpam-5048	261	10	a	a	DET
ejpam-5048	261	11	qge	qge	NOUN
ejpam-5048	261	12	-	-	NOUN
ejpam-5048	261	13	algebra	algebra	NOUN
ejpam-5048	261	14	and	and	CCONJ
ejpam-5048	261	15	f	f	NOUN
ejpam-5048	261	16	=	=	NOUN
ejpam-5048	261	17	{	{	PUNCT
ejpam-5048	261	18	1	1	NUM
ejpam-5048	261	19	,	,	PUNCT
ejpam-5048	261	20	b	b	NOUN
ejpam-5048	261	21	,	,	PUNCT
ejpam-5048	261	22	c	c	NOUN
ejpam-5048	261	23	}	}	PUNCT
ejpam-5048	261	24	a	a	DET
ejpam-5048	261	25	qge	qge	NOUN
ejpam-5048	261	26	-	-	NOUN
ejpam-5048	261	27	filter	filter	NOUN
ejpam-5048	261	28	of	of	ADP
ejpam-5048	261	29	x	x	PUNCT
ejpam-5048	261	30	given	give	VERB
ejpam-5048	261	31	in	in	ADP
ejpam-5048	261	32	example	example	NOUN
ejpam-5048	261	33	16	16	NUM
ejpam-5048	261	34	.	.	PUNCT
ejpam-5048	262	1	then	then	ADV
ejpam-5048	262	2	f	f	PROPN
ejpam-5048	262	3	does	do	AUX
ejpam-5048	262	4	not	not	PART
ejpam-5048	262	5	satisfy	satisfy	VERB
ejpam-5048	262	6	(	(	PUNCT
ejpam-5048	262	7	26	26	NUM
ejpam-5048	262	8	)	)	PUNCT
ejpam-5048	262	9	since	since	SCONJ
ejpam-5048	262	10	d∗(c∗e	d∗(c∗e	ADJ
ejpam-5048	262	11	)	)	PUNCT
ejpam-5048	262	12	=	=	PUNCT
ejpam-5048	263	1	d∗d	d∗d	PROPN
ejpam-5048	263	2	=	=	SYM
ejpam-5048	263	3	1	1	NUM
ejpam-5048	263	4	∈	∈	NOUN
ejpam-5048	263	5	f	f	NOUN
ejpam-5048	263	6	but	but	CCONJ
ejpam-5048	263	7	c∗e	c∗e	NOUN
ejpam-5048	263	8	=	=	SYM
ejpam-5048	263	9	d	d	PROPN
ejpam-5048	263	10	/∈	/∈	PUNCT
ejpam-5048	264	1	f	f	PROPN
ejpam-5048	264	2	.	.	PUNCT
ejpam-5048	265	1	we	we	PRON
ejpam-5048	265	2	explore	explore	VERB
ejpam-5048	265	3	the	the	DET
ejpam-5048	265	4	conditions	condition	NOUN
ejpam-5048	265	5	for	for	ADP
ejpam-5048	265	6	a	a	DET
ejpam-5048	265	7	qge	qge	NOUN
ejpam-5048	265	8	-	-	NOUN
ejpam-5048	265	9	filter	filter	NOUN
ejpam-5048	265	10	to	to	PART
ejpam-5048	265	11	satisfy	satisfy	VERB
ejpam-5048	265	12	the	the	DET
ejpam-5048	265	13	condition	condition	NOUN
ejpam-5048	265	14	(	(	PUNCT
ejpam-5048	265	15	25	25	NUM
ejpam-5048	265	16	)	)	PUNCT
ejpam-5048	265	17	.	.	PUNCT
ejpam-5048	266	1	theorem	theorem	ADJ
ejpam-5048	266	2	7	7	NUM
ejpam-5048	266	3	.	.	PUNCT
ejpam-5048	267	1	let	let	VERB
ejpam-5048	267	2	f	f	PRON
ejpam-5048	267	3	be	be	AUX
ejpam-5048	267	4	a	a	DET
ejpam-5048	267	5	qge	qge	NOUN
ejpam-5048	267	6	-	-	NOUN
ejpam-5048	267	7	filter	filter	NOUN
ejpam-5048	267	8	of	of	ADP
ejpam-5048	267	9	x.	x.	NOUN
ejpam-5048	267	10	if	if	SCONJ
ejpam-5048	267	11	f	f	PROPN
ejpam-5048	267	12	satisfies	satisfy	VERB
ejpam-5048	267	13	(	(	PUNCT
ejpam-5048	267	14	26	26	NUM
ejpam-5048	267	15	)	)	PUNCT
ejpam-5048	267	16	,	,	PUNCT
ejpam-5048	267	17	then	then	ADV
ejpam-5048	267	18	it	it	PRON
ejpam-5048	267	19	satisfies	satisfy	VERB
ejpam-5048	267	20	the	the	DET
ejpam-5048	267	21	condition	condition	NOUN
ejpam-5048	267	22	(	(	PUNCT
ejpam-5048	267	23	25	25	NUM
ejpam-5048	267	24	)	)	PUNCT
ejpam-5048	267	25	.	.	PUNCT
ejpam-5048	268	1	y.	y.	PROPN
ejpam-5048	268	2	b.	b.	PROPN
ejpam-5048	268	3	jun	jun	PROPN
ejpam-5048	268	4	,	,	PUNCT
ejpam-5048	268	5	ravikumar	ravikumar	PROPN
ejpam-5048	268	6	bandaru	bandaru	PROPN
ejpam-5048	268	7	,	,	PUNCT
ejpam-5048	268	8	rahul	rahul	PROPN
ejpam-5048	268	9	shukla	shukla	PROPN
ejpam-5048	268	10	/	/	SYM
ejpam-5048	268	11	eur	eur	PROPN
ejpam-5048	268	12	.	.	PUNCT
ejpam-5048	269	1	j.	j.	PROPN
ejpam-5048	269	2	pure	pure	PROPN
ejpam-5048	269	3	appl	appl	PROPN
ejpam-5048	269	4	.	.	PROPN
ejpam-5048	269	5	math	math	PROPN
ejpam-5048	269	6	,	,	PUNCT
ejpam-5048	269	7	17	17	NUM
ejpam-5048	269	8	(	(	PUNCT
ejpam-5048	269	9	1	1	NUM
ejpam-5048	269	10	)	)	PUNCT
ejpam-5048	269	11	(	(	PUNCT
ejpam-5048	269	12	2024	2024	NUM
ejpam-5048	269	13	)	)	PUNCT
ejpam-5048	269	14	,	,	PUNCT
ejpam-5048	269	15	569	569	NUM
ejpam-5048	269	16	-	-	SYM
ejpam-5048	269	17	581	581	NUM
ejpam-5048	269	18	578	578	NUM
ejpam-5048	269	19	proof	proof	NOUN
ejpam-5048	269	20	.	.	PUNCT
ejpam-5048	270	1	let	let	VERB
ejpam-5048	270	2	f	f	PRON
ejpam-5048	270	3	be	be	AUX
ejpam-5048	270	4	a	a	DET
ejpam-5048	270	5	qge	qge	NOUN
ejpam-5048	270	6	-	-	NOUN
ejpam-5048	270	7	filter	filter	NOUN
ejpam-5048	270	8	of	of	ADP
ejpam-5048	270	9	x	x	PRON
ejpam-5048	270	10	that	that	SCONJ
ejpam-5048	270	11	satisfies	satisfie	NOUN
ejpam-5048	270	12	(	(	PUNCT
ejpam-5048	270	13	26	26	NUM
ejpam-5048	270	14	)	)	PUNCT
ejpam-5048	270	15	.	.	PUNCT
ejpam-5048	271	1	let	let	VERB
ejpam-5048	271	2	κ	κ	NOUN
ejpam-5048	271	3	,	,	PUNCT
ejpam-5048	271	4	δ	δ	PROPN
ejpam-5048	271	5	,	,	PUNCT
ejpam-5048	271	6	ς	ς	PROPN
ejpam-5048	271	7	∈	∈	PROPN
ejpam-5048	271	8	x	x	AUX
ejpam-5048	271	9	be	be	AUX
ejpam-5048	271	10	such	such	ADJ
ejpam-5048	271	11	that	that	SCONJ
ejpam-5048	271	12	ς	ς	PROPN
ejpam-5048	271	13	∗	∗	NOUN
ejpam-5048	271	14	(	(	PUNCT
ejpam-5048	271	15	δ	δ	PROPN
ejpam-5048	271	16	∗	∗	PROPN
ejpam-5048	271	17	κ	κ	NOUN
ejpam-5048	271	18	)	)	PUNCT
ejpam-5048	271	19	∈	∈	PROPN
ejpam-5048	271	20	f	f	PROPN
ejpam-5048	271	21	and	and	CCONJ
ejpam-5048	271	22	ς	ς	PROPN
ejpam-5048	272	1	∗	∗	NOUN
ejpam-5048	272	2	δ	δ	NOUN
ejpam-5048	272	3	∈	∈	PROPN
ejpam-5048	272	4	f	f	PROPN
ejpam-5048	272	5	.	.	PUNCT
ejpam-5048	273	1	then	then	ADV
ejpam-5048	273	2	δ	δ	PROPN
ejpam-5048	273	3	∗	∗	VERB
ejpam-5048	273	4	κ	κ	PROPN
ejpam-5048	273	5	∈	∈	PROPN
ejpam-5048	273	6	f	f	PROPN
ejpam-5048	273	7	and	and	CCONJ
ejpam-5048	273	8	ς	ς	PROPN
ejpam-5048	273	9	∗	∗	NOUN
ejpam-5048	273	10	δ	δ	NOUN
ejpam-5048	274	1	∈	∈	PROPN
ejpam-5048	274	2	f	f	X
ejpam-5048	274	3	.	.	PUNCT
ejpam-5048	275	1	it	it	PRON
ejpam-5048	275	2	follows	follow	VERB
ejpam-5048	275	3	from	from	ADP
ejpam-5048	275	4	(	(	PUNCT
ejpam-5048	275	5	10	10	NUM
ejpam-5048	275	6	)	)	PUNCT
ejpam-5048	275	7	that	that	SCONJ
ejpam-5048	275	8	(	(	PUNCT
ejpam-5048	275	9	ς	ς	PROPN
ejpam-5048	275	10	∗	∗	X
ejpam-5048	275	11	δ	δ	PROPN
ejpam-5048	275	12	)	)	PUNCT
ejpam-5048	275	13	∗	∗	NOUN
ejpam-5048	275	14	(	(	PUNCT
ejpam-5048	275	15	ς	ς	PROPN
ejpam-5048	275	16	∗	∗	NOUN
ejpam-5048	275	17	κ	κ	NOUN
ejpam-5048	275	18	)	)	PUNCT
ejpam-5048	276	1	=	=	SYM
ejpam-5048	276	2	δ	δ	PROPN
ejpam-5048	276	3	∗	∗	VERB
ejpam-5048	276	4	κ	κ	X
ejpam-5048	276	5	∈	∈	PROPN
ejpam-5048	276	6	f	f	PROPN
ejpam-5048	276	7	and	and	CCONJ
ejpam-5048	276	8	ς	ς	PROPN
ejpam-5048	276	9	∗	∗	NOUN
ejpam-5048	276	10	δ	δ	NOUN
ejpam-5048	277	1	∈	∈	PROPN
ejpam-5048	277	2	f	f	PROPN
ejpam-5048	277	3	.	.	PUNCT
ejpam-5048	278	1	hence	hence	ADV
ejpam-5048	278	2	ς	ς	PROPN
ejpam-5048	278	3	∗	∗	NOUN
ejpam-5048	278	4	κ	κ	PRON
ejpam-5048	278	5	∈	∈	PROPN
ejpam-5048	278	6	f	f	X
ejpam-5048	278	7	by	by	ADP
ejpam-5048	278	8	(	(	PUNCT
ejpam-5048	278	9	23	23	NUM
ejpam-5048	278	10	)	)	PUNCT
ejpam-5048	278	11	,	,	PUNCT
ejpam-5048	278	12	and	and	CCONJ
ejpam-5048	278	13	therefore	therefore	ADV
ejpam-5048	278	14	the	the	DET
ejpam-5048	278	15	condition	condition	NOUN
ejpam-5048	278	16	(	(	PUNCT
ejpam-5048	278	17	25	25	NUM
ejpam-5048	278	18	)	)	PUNCT
ejpam-5048	278	19	is	be	AUX
ejpam-5048	278	20	valid	valid	ADJ
ejpam-5048	278	21	.	.	PUNCT
ejpam-5048	279	1	we	we	PRON
ejpam-5048	279	2	explore	explore	VERB
ejpam-5048	279	3	the	the	DET
ejpam-5048	279	4	conditions	condition	NOUN
ejpam-5048	279	5	for	for	ADP
ejpam-5048	279	6	a	a	DET
ejpam-5048	279	7	subset	subset	NOUN
ejpam-5048	279	8	f	f	PROPN
ejpam-5048	279	9	of	of	ADP
ejpam-5048	279	10	x	x	INTJ
ejpam-5048	279	11	to	to	PART
ejpam-5048	279	12	be	be	AUX
ejpam-5048	279	13	a	a	DET
ejpam-5048	279	14	qge	qge	NOUN
ejpam-5048	279	15	-	-	NOUN
ejpam-5048	279	16	filter	filter	NOUN
ejpam-5048	279	17	of	of	ADP
ejpam-5048	279	18	x.	x.	PROPN
ejpam-5048	279	19	theorem	theorem	VERB
ejpam-5048	279	20	8	8	NUM
ejpam-5048	279	21	.	.	PUNCT
ejpam-5048	280	1	let	let	VERB
ejpam-5048	280	2	f	f	PRON
ejpam-5048	280	3	be	be	AUX
ejpam-5048	280	4	a	a	DET
ejpam-5048	280	5	subset	subset	NOUN
ejpam-5048	280	6	of	of	ADP
ejpam-5048	280	7	x	x	PRON
ejpam-5048	280	8	which	which	PRON
ejpam-5048	280	9	includes	include	VERB
ejpam-5048	280	10	the	the	DET
ejpam-5048	280	11	unit	unit	NOUN
ejpam-5048	280	12	1	1	NUM
ejpam-5048	280	13	.	.	PUNCT
ejpam-5048	281	1	if	if	SCONJ
ejpam-5048	281	2	f	f	PROPN
ejpam-5048	281	3	satisfies	satisfy	VERB
ejpam-5048	281	4	the	the	DET
ejpam-5048	281	5	condition	condition	NOUN
ejpam-5048	281	6	(	(	PUNCT
ejpam-5048	281	7	26	26	NUM
ejpam-5048	281	8	)	)	PUNCT
ejpam-5048	281	9	,	,	PUNCT
ejpam-5048	281	10	then	then	ADV
ejpam-5048	281	11	f	f	PROPN
ejpam-5048	281	12	is	be	AUX
ejpam-5048	281	13	a	a	DET
ejpam-5048	281	14	qge	qge	NOUN
ejpam-5048	281	15	-	-	NOUN
ejpam-5048	281	16	filter	filter	NOUN
ejpam-5048	281	17	of	of	ADP
ejpam-5048	281	18	x.	x.	NOUN
ejpam-5048	281	19	proof	proof	PROPN
ejpam-5048	281	20	.	.	PUNCT
ejpam-5048	282	1	assume	assume	VERB
ejpam-5048	282	2	that	that	SCONJ
ejpam-5048	282	3	a	a	DET
ejpam-5048	282	4	subset	subset	NOUN
ejpam-5048	282	5	f	f	X
ejpam-5048	282	6	of	of	ADP
ejpam-5048	282	7	x	x	PUNCT
ejpam-5048	282	8	includes	include	VERB
ejpam-5048	282	9	the	the	DET
ejpam-5048	282	10	unit	unit	NOUN
ejpam-5048	282	11	1	1	NUM
ejpam-5048	282	12	and	and	CCONJ
ejpam-5048	282	13	satisfies	satisfy	VERB
ejpam-5048	282	14	the	the	DET
ejpam-5048	282	15	condition	condition	NOUN
ejpam-5048	282	16	(	(	PUNCT
ejpam-5048	282	17	26	26	NUM
ejpam-5048	282	18	)	)	PUNCT
ejpam-5048	282	19	.	.	PUNCT
ejpam-5048	283	1	let	let	VERB
ejpam-5048	283	2	κ	κ	VERB
ejpam-5048	283	3	,	,	PUNCT
ejpam-5048	283	4	δ	δ	PROPN
ejpam-5048	283	5	∈	∈	PROPN
ejpam-5048	283	6	x	x	AUX
ejpam-5048	283	7	be	be	AUX
ejpam-5048	283	8	such	such	ADJ
ejpam-5048	283	9	that	that	SCONJ
ejpam-5048	283	10	κ	κ	PROPN
ejpam-5048	283	11	∗	∗	NOUN
ejpam-5048	283	12	δ	δ	PROPN
ejpam-5048	283	13	∈	∈	PROPN
ejpam-5048	283	14	f	f	PROPN
ejpam-5048	283	15	and	and	CCONJ
ejpam-5048	283	16	κ	κ	PROPN
ejpam-5048	283	17	∈	∈	PROPN
ejpam-5048	284	1	f	f	X
ejpam-5048	284	2	.	.	PUNCT
ejpam-5048	285	1	then	then	ADV
ejpam-5048	285	2	κ	κ	X
ejpam-5048	285	3	∗	∗	NOUN
ejpam-5048	285	4	(	(	PUNCT
ejpam-5048	285	5	1	1	NUM
ejpam-5048	285	6	∗	∗	NUM
ejpam-5048	285	7	δ	δ	PROPN
ejpam-5048	285	8	)	)	PUNCT
ejpam-5048	286	1	=	=	SYM
ejpam-5048	286	2	κ	κ	NOUN
ejpam-5048	286	3	∗	∗	X
ejpam-5048	286	4	δ	δ	PROPN
ejpam-5048	286	5	∈	∈	PROPN
ejpam-5048	286	6	f	f	X
ejpam-5048	286	7	by	by	ADP
ejpam-5048	286	8	(	(	PUNCT
ejpam-5048	286	9	ge2	ge2	NOUN
ejpam-5048	286	10	)	)	PUNCT
ejpam-5048	286	11	.	.	PUNCT
ejpam-5048	287	1	it	it	PRON
ejpam-5048	287	2	follows	follow	VERB
ejpam-5048	287	3	from	from	ADP
ejpam-5048	287	4	(	(	PUNCT
ejpam-5048	287	5	ge2	ge2	NOUN
ejpam-5048	287	6	)	)	PUNCT
ejpam-5048	287	7	and	and	CCONJ
ejpam-5048	287	8	(	(	PUNCT
ejpam-5048	287	9	26	26	NUM
ejpam-5048	287	10	)	)	PUNCT
ejpam-5048	287	11	that	that	SCONJ
ejpam-5048	287	12	δ	δ	X
ejpam-5048	287	13	=	=	SYM
ejpam-5048	287	14	1	1	NUM
ejpam-5048	287	15	∗	∗	NOUN
ejpam-5048	287	16	δ	δ	NOUN
ejpam-5048	287	17	∈	∈	PROPN
ejpam-5048	287	18	f	f	PROPN
ejpam-5048	287	19	.	.	PUNCT
ejpam-5048	288	1	hence	hence	ADV
ejpam-5048	288	2	f	f	PROPN
ejpam-5048	288	3	is	be	AUX
ejpam-5048	288	4	a	a	DET
ejpam-5048	288	5	qge	qge	NOUN
ejpam-5048	288	6	-	-	NOUN
ejpam-5048	288	7	filter	filter	NOUN
ejpam-5048	288	8	of	of	ADP
ejpam-5048	288	9	x.	x.	NOUN
ejpam-5048	288	10	theorem	theorem	VERB
ejpam-5048	288	11	9	9	NUM
ejpam-5048	288	12	.	.	PUNCT
ejpam-5048	289	1	let	let	VERB
ejpam-5048	289	2	f	f	PRON
ejpam-5048	289	3	be	be	AUX
ejpam-5048	289	4	a	a	DET
ejpam-5048	289	5	subset	subset	NOUN
ejpam-5048	289	6	of	of	ADP
ejpam-5048	289	7	x	x	PUNCT
ejpam-5048	289	8	with	with	ADP
ejpam-5048	289	9	the	the	DET
ejpam-5048	289	10	unit	unit	NOUN
ejpam-5048	289	11	1	1	NUM
ejpam-5048	289	12	.	.	PUNCT
ejpam-5048	290	1	if	if	SCONJ
ejpam-5048	290	2	it	it	PRON
ejpam-5048	290	3	satisfies	satisfy	VERB
ejpam-5048	290	4	:	:	PUNCT
ejpam-5048	290	5	(	(	PUNCT
ejpam-5048	290	6	∀κ	∀κ	ADV
ejpam-5048	290	7	,	,	PUNCT
ejpam-5048	290	8	δ	δ	PROPN
ejpam-5048	290	9	,	,	PUNCT
ejpam-5048	290	10	ς	ς	PROPN
ejpam-5048	290	11	∈	∈	PROPN
ejpam-5048	290	12	x)(κ	x)(κ	PROPN
ejpam-5048	290	13	∗	∗	NOUN
ejpam-5048	290	14	(	(	PUNCT
ejpam-5048	290	15	δ	δ	PROPN
ejpam-5048	290	16	∗	∗	PROPN
ejpam-5048	290	17	ς	ς	NOUN
ejpam-5048	290	18	)	)	PUNCT
ejpam-5048	290	19	∈	∈	PROPN
ejpam-5048	290	20	f	f	PROPN
ejpam-5048	290	21	,	,	PUNCT
ejpam-5048	290	22	δ	δ	PROPN
ejpam-5048	290	23	∈	∈	PROPN
ejpam-5048	290	24	f	f	PROPN
ejpam-5048	290	25	⇒	⇒	VERB
ejpam-5048	290	26	κ	κ	PROPN
ejpam-5048	290	27	∗	∗	NOUN
ejpam-5048	290	28	ς	ς	PROPN
ejpam-5048	290	29	∈	∈	PROPN
ejpam-5048	290	30	f	f	PROPN
ejpam-5048	290	31	)	)	PUNCT
ejpam-5048	290	32	,	,	PUNCT
ejpam-5048	290	33	(	(	PUNCT
ejpam-5048	290	34	27	27	NUM
ejpam-5048	290	35	)	)	PUNCT
ejpam-5048	290	36	then	then	ADV
ejpam-5048	290	37	it	it	PRON
ejpam-5048	290	38	is	be	AUX
ejpam-5048	290	39	a	a	DET
ejpam-5048	290	40	qge	qge	NOUN
ejpam-5048	290	41	-	-	NOUN
ejpam-5048	290	42	filter	filter	NOUN
ejpam-5048	290	43	of	of	ADP
ejpam-5048	290	44	x.	x.	NOUN
ejpam-5048	290	45	proof	proof	NOUN
ejpam-5048	290	46	.	.	PUNCT
ejpam-5048	291	1	let	let	VERB
ejpam-5048	291	2	κ	κ	VERB
ejpam-5048	291	3	,	,	PUNCT
ejpam-5048	291	4	δ	δ	PROPN
ejpam-5048	291	5	∈	∈	PROPN
ejpam-5048	291	6	x	x	AUX
ejpam-5048	291	7	be	be	AUX
ejpam-5048	291	8	such	such	ADJ
ejpam-5048	291	9	that	that	SCONJ
ejpam-5048	291	10	κ	κ	PROPN
ejpam-5048	291	11	∗	∗	NOUN
ejpam-5048	291	12	δ	δ	PROPN
ejpam-5048	291	13	∈	∈	PROPN
ejpam-5048	291	14	f	f	PROPN
ejpam-5048	291	15	and	and	CCONJ
ejpam-5048	291	16	κ	κ	PROPN
ejpam-5048	291	17	∈	∈	PROPN
ejpam-5048	291	18	f	f	X
ejpam-5048	291	19	.	.	PUNCT
ejpam-5048	292	1	using	use	VERB
ejpam-5048	292	2	(	(	PUNCT
ejpam-5048	292	3	ge2	ge2	NOUN
ejpam-5048	292	4	)	)	PUNCT
ejpam-5048	292	5	,	,	PUNCT
ejpam-5048	292	6	we	we	PRON
ejpam-5048	292	7	have	have	VERB
ejpam-5048	292	8	1	1	NUM
ejpam-5048	292	9	∗	∗	NOUN
ejpam-5048	292	10	(	(	PUNCT
ejpam-5048	292	11	κ	κ	NOUN
ejpam-5048	292	12	∗	∗	X
ejpam-5048	292	13	δ	δ	PROPN
ejpam-5048	292	14	)	)	PUNCT
ejpam-5048	293	1	=	=	SYM
ejpam-5048	293	2	κ	κ	NOUN
ejpam-5048	293	3	∗	∗	X
ejpam-5048	293	4	δ	δ	PROPN
ejpam-5048	293	5	∈	∈	PROPN
ejpam-5048	293	6	f	f	X
ejpam-5048	293	7	,	,	PUNCT
ejpam-5048	293	8	and	and	CCONJ
ejpam-5048	293	9	so	so	ADV
ejpam-5048	293	10	δ	δ	NOUN
ejpam-5048	293	11	=	=	SYM
ejpam-5048	293	12	1	1	NUM
ejpam-5048	293	13	∗	∗	NOUN
ejpam-5048	293	14	δ	δ	NOUN
ejpam-5048	293	15	∈	∈	PROPN
ejpam-5048	293	16	f	f	X
ejpam-5048	293	17	by	by	ADP
ejpam-5048	293	18	(	(	PUNCT
ejpam-5048	293	19	ge2	ge2	PROPN
ejpam-5048	293	20	)	)	PUNCT
ejpam-5048	293	21	and	and	CCONJ
ejpam-5048	293	22	(	(	PUNCT
ejpam-5048	293	23	27	27	NUM
ejpam-5048	293	24	)	)	PUNCT
ejpam-5048	293	25	.	.	PUNCT
ejpam-5048	294	1	hence	hence	ADV
ejpam-5048	294	2	f	f	PROPN
ejpam-5048	294	3	is	be	AUX
ejpam-5048	294	4	a	a	DET
ejpam-5048	294	5	qge	qge	NOUN
ejpam-5048	294	6	-	-	NOUN
ejpam-5048	294	7	filter	filter	NOUN
ejpam-5048	294	8	of	of	ADP
ejpam-5048	294	9	x.	x.	NOUN
ejpam-5048	294	10	in	in	ADP
ejpam-5048	294	11	the	the	DET
ejpam-5048	294	12	following	follow	VERB
ejpam-5048	294	13	example	example	NOUN
ejpam-5048	294	14	,	,	PUNCT
ejpam-5048	294	15	we	we	PRON
ejpam-5048	294	16	can	can	AUX
ejpam-5048	294	17	find	find	VERB
ejpam-5048	294	18	a	a	DET
ejpam-5048	294	19	qge	qge	NOUN
ejpam-5048	294	20	-	-	NOUN
ejpam-5048	294	21	filter	filter	NOUN
ejpam-5048	294	22	of	of	ADP
ejpam-5048	294	23	x	x	PRON
ejpam-5048	294	24	which	which	PRON
ejpam-5048	294	25	does	do	AUX
ejpam-5048	294	26	not	not	PART
ejpam-5048	294	27	satisfy	satisfy	VERB
ejpam-5048	294	28	the	the	DET
ejpam-5048	294	29	condition	condition	NOUN
ejpam-5048	294	30	(	(	PUNCT
ejpam-5048	294	31	27	27	NUM
ejpam-5048	294	32	)	)	PUNCT
ejpam-5048	294	33	.	.	PUNCT
ejpam-5048	295	1	example	example	NOUN
ejpam-5048	296	1	18	18	NUM
ejpam-5048	296	2	.	.	PUNCT
ejpam-5048	296	3	consider	consider	VERB
ejpam-5048	296	4	the	the	DET
ejpam-5048	296	5	qge	qge	NOUN
ejpam-5048	296	6	-	-	NOUN
ejpam-5048	296	7	algebra	algebra	NOUN
ejpam-5048	296	8	(	(	PUNCT
ejpam-5048	296	9	x	x	X
ejpam-5048	296	10	,	,	PUNCT
ejpam-5048	296	11	∗	∗	NOUN
ejpam-5048	296	12	,	,	PUNCT
ejpam-5048	296	13	1	1	NUM
ejpam-5048	296	14	)	)	PUNCT
ejpam-5048	296	15	given	give	VERB
ejpam-5048	296	16	in	in	ADP
ejpam-5048	296	17	example	example	NOUN
ejpam-5048	296	18	13	13	NUM
ejpam-5048	296	19	.	.	PUNCT
ejpam-5048	297	1	it	it	PRON
ejpam-5048	297	2	is	be	AUX
ejpam-5048	297	3	routine	routine	ADJ
ejpam-5048	297	4	to	to	PART
ejpam-5048	297	5	verify	verify	VERB
ejpam-5048	297	6	that	that	SCONJ
ejpam-5048	297	7	the	the	DET
ejpam-5048	297	8	set	set	NOUN
ejpam-5048	297	9	f	f	NOUN
ejpam-5048	297	10	:	:	PUNCT
ejpam-5048	297	11	=	=	SYM
ejpam-5048	297	12	{	{	PUNCT
ejpam-5048	297	13	1	1	NUM
ejpam-5048	297	14	,	,	PUNCT
ejpam-5048	297	15	e	e	NOUN
ejpam-5048	297	16	}	}	PUNCT
ejpam-5048	297	17	is	be	AUX
ejpam-5048	297	18	a	a	DET
ejpam-5048	297	19	qge	qge	NOUN
ejpam-5048	297	20	-	-	NOUN
ejpam-5048	297	21	filter	filter	NOUN
ejpam-5048	297	22	of	of	ADP
ejpam-5048	297	23	x.	x.	NOUN
ejpam-5048	298	1	but	but	CCONJ
ejpam-5048	298	2	f	f	PROPN
ejpam-5048	298	3	does	do	AUX
ejpam-5048	298	4	not	not	PART
ejpam-5048	298	5	satisfy	satisfy	VERB
ejpam-5048	298	6	(	(	PUNCT
ejpam-5048	298	7	27	27	NUM
ejpam-5048	298	8	)	)	PUNCT
ejpam-5048	298	9	since	since	SCONJ
ejpam-5048	298	10	a	a	DET
ejpam-5048	298	11	∗	∗	NOUN
ejpam-5048	298	12	(	(	PUNCT
ejpam-5048	298	13	e	e	NOUN
ejpam-5048	298	14	∗	∗	X
ejpam-5048	298	15	b	b	NOUN
ejpam-5048	298	16	)	)	PUNCT
ejpam-5048	298	17	=	=	PUNCT
ejpam-5048	298	18	a	a	DET
ejpam-5048	298	19	∗	∗	NOUN
ejpam-5048	298	20	d	d	NOUN
ejpam-5048	298	21	=	=	SYM
ejpam-5048	298	22	e	e	PROPN
ejpam-5048	298	23	∈	∈	PROPN
ejpam-5048	298	24	f	f	PROPN
ejpam-5048	298	25	and	and	CCONJ
ejpam-5048	298	26	e	e	PROPN
ejpam-5048	298	27	∈	∈	PROPN
ejpam-5048	298	28	f	f	PROPN
ejpam-5048	298	29	but	but	CCONJ
ejpam-5048	298	30	a	a	DET
ejpam-5048	298	31	∗	∗	NOUN
ejpam-5048	298	32	b	b	NOUN
ejpam-5048	299	1	=	=	SYM
ejpam-5048	299	2	c	c	PROPN
ejpam-5048	299	3	/∈	/∈	PUNCT
ejpam-5048	300	1	f.	f.	PROPN
ejpam-5048	300	2	definition	definition	NOUN
ejpam-5048	300	3	6	6	NUM
ejpam-5048	300	4	.	.	PUNCT
ejpam-5048	301	1	if	if	SCONJ
ejpam-5048	301	2	a	a	DET
ejpam-5048	301	3	subset	subset	NOUN
ejpam-5048	301	4	f	f	X
ejpam-5048	301	5	of	of	ADP
ejpam-5048	301	6	x	x	PUNCT
ejpam-5048	301	7	satisfies	satisfie	NOUN
ejpam-5048	301	8	(	(	PUNCT
ejpam-5048	301	9	22	22	NUM
ejpam-5048	301	10	)	)	PUNCT
ejpam-5048	301	11	and	and	CCONJ
ejpam-5048	301	12	(	(	PUNCT
ejpam-5048	301	13	27	27	NUM
ejpam-5048	301	14	)	)	PUNCT
ejpam-5048	301	15	,	,	PUNCT
ejpam-5048	301	16	we	we	PRON
ejpam-5048	301	17	say	say	VERB
ejpam-5048	301	18	that	that	SCONJ
ejpam-5048	301	19	f	f	PROPN
ejpam-5048	301	20	is	be	AUX
ejpam-5048	301	21	a	a	DET
ejpam-5048	301	22	strong	strong	ADJ
ejpam-5048	301	23	qge	qge	NOUN
ejpam-5048	301	24	-	-	NOUN
ejpam-5048	301	25	filter	filter	NOUN
ejpam-5048	301	26	of	of	ADP
ejpam-5048	301	27	x.	x.	PROPN
ejpam-5048	301	28	example	example	NOUN
ejpam-5048	302	1	19	19	NUM
ejpam-5048	302	2	.	.	PUNCT
ejpam-5048	303	1	let	let	VERB
ejpam-5048	303	2	x	x	PUNCT
ejpam-5048	303	3	=	=	PUNCT
ejpam-5048	303	4	{	{	PUNCT
ejpam-5048	303	5	1	1	NUM
ejpam-5048	303	6	,	,	PUNCT
ejpam-5048	303	7	a	a	DET
ejpam-5048	303	8	,	,	PUNCT
ejpam-5048	303	9	b	b	NOUN
ejpam-5048	303	10	,	,	PUNCT
ejpam-5048	303	11	c	c	AUX
ejpam-5048	303	12	}	}	PUNCT
ejpam-5048	303	13	be	be	AUX
ejpam-5048	303	14	a	a	DET
ejpam-5048	303	15	set	set	NOUN
ejpam-5048	303	16	with	with	ADP
ejpam-5048	303	17	a	a	DET
ejpam-5048	303	18	binary	binary	ADJ
ejpam-5048	303	19	operation	operation	NOUN
ejpam-5048	303	20	“	"	PUNCT
ejpam-5048	303	21	∗	∗	NOUN
ejpam-5048	303	22	”	"	PUNCT
ejpam-5048	303	23	given	give	VERB
ejpam-5048	303	24	in	in	ADP
ejpam-5048	303	25	the	the	DET
ejpam-5048	303	26	following	follow	VERB
ejpam-5048	303	27	table	table	NOUN
ejpam-5048	303	28	:	:	PUNCT
ejpam-5048	303	29	∗	∗	NOUN
ejpam-5048	303	30	1	1	NUM
ejpam-5048	303	31	a	a	DET
ejpam-5048	303	32	b	b	NOUN
ejpam-5048	303	33	c	c	NOUN
ejpam-5048	303	34	1	1	NUM
ejpam-5048	303	35	1	1	NUM
ejpam-5048	303	36	a	a	DET
ejpam-5048	303	37	b	b	NOUN
ejpam-5048	303	38	c	c	NOUN
ejpam-5048	303	39	a	a	DET
ejpam-5048	303	40	b	b	NUM
ejpam-5048	303	41	1	1	NUM
ejpam-5048	303	42	c	c	PROPN
ejpam-5048	303	43	a	a	DET
ejpam-5048	303	44	b	b	NOUN
ejpam-5048	303	45	a	a	PRON
ejpam-5048	303	46	c	c	NOUN
ejpam-5048	303	47	1	1	NUM
ejpam-5048	303	48	b	b	NOUN
ejpam-5048	303	49	c	c	NOUN
ejpam-5048	303	50	c	c	PROPN
ejpam-5048	303	51	b	b	PROPN
ejpam-5048	303	52	a	a	DET
ejpam-5048	303	53	1	1	NUM
ejpam-5048	303	54	then	then	ADV
ejpam-5048	303	55	(	(	PUNCT
ejpam-5048	303	56	x	x	X
ejpam-5048	303	57	,	,	PUNCT
ejpam-5048	303	58	∗	∗	NOUN
ejpam-5048	303	59	,	,	PUNCT
ejpam-5048	303	60	1	1	NUM
ejpam-5048	303	61	)	)	PUNCT
ejpam-5048	303	62	is	be	AUX
ejpam-5048	303	63	a	a	DET
ejpam-5048	303	64	qge	qge	NOUN
ejpam-5048	303	65	-	-	NOUN
ejpam-5048	303	66	algebra	algebra	NOUN
ejpam-5048	303	67	,	,	PUNCT
ejpam-5048	303	68	and	and	CCONJ
ejpam-5048	303	69	the	the	DET
ejpam-5048	303	70	set	set	NOUN
ejpam-5048	303	71	f	f	NOUN
ejpam-5048	303	72	:	:	PUNCT
ejpam-5048	303	73	=	=	SYM
ejpam-5048	303	74	{	{	PUNCT
ejpam-5048	303	75	1	1	NUM
ejpam-5048	303	76	,	,	PUNCT
ejpam-5048	303	77	c	c	NOUN
ejpam-5048	303	78	}	}	PUNCT
ejpam-5048	303	79	is	be	AUX
ejpam-5048	303	80	a	a	DET
ejpam-5048	303	81	strong	strong	ADJ
ejpam-5048	303	82	qge	qge	NOUN
ejpam-5048	303	83	-	-	NOUN
ejpam-5048	303	84	filter	filter	NOUN
ejpam-5048	303	85	of	of	ADP
ejpam-5048	303	86	x.	x.	NOUN
ejpam-5048	303	87	it	it	PRON
ejpam-5048	303	88	is	be	AUX
ejpam-5048	303	89	obvious	obvious	ADJ
ejpam-5048	303	90	that	that	SCONJ
ejpam-5048	303	91	every	every	DET
ejpam-5048	303	92	strong	strong	ADJ
ejpam-5048	303	93	qge	qge	NOUN
ejpam-5048	303	94	-	-	NOUN
ejpam-5048	303	95	filter	filter	NOUN
ejpam-5048	303	96	is	be	AUX
ejpam-5048	303	97	a	a	DET
ejpam-5048	303	98	qge	qge	NOUN
ejpam-5048	303	99	-	-	NOUN
ejpam-5048	303	100	filter	filter	NOUN
ejpam-5048	303	101	(	(	PUNCT
ejpam-5048	303	102	see	see	VERB
ejpam-5048	303	103	theorem	theorem	NOUN
ejpam-5048	303	104	9	9	NUM
ejpam-5048	303	105	)	)	PUNCT
ejpam-5048	303	106	.	.	PUNCT
ejpam-5048	304	1	but	but	CCONJ
ejpam-5048	304	2	a	a	DET
ejpam-5048	304	3	qge	qge	NOUN
ejpam-5048	304	4	-	-	NOUN
ejpam-5048	304	5	filter	filter	NOUN
ejpam-5048	304	6	may	may	AUX
ejpam-5048	304	7	not	not	PART
ejpam-5048	304	8	be	be	AUX
ejpam-5048	304	9	a	a	DET
ejpam-5048	304	10	strong	strong	ADJ
ejpam-5048	304	11	qge	qge	NOUN
ejpam-5048	304	12	-	-	NOUN
ejpam-5048	304	13	filter	filter	NOUN
ejpam-5048	304	14	as	as	SCONJ
ejpam-5048	304	15	seen	see	VERB
ejpam-5048	304	16	in	in	ADP
ejpam-5048	304	17	the	the	DET
ejpam-5048	304	18	following	follow	VERB
ejpam-5048	304	19	example	example	NOUN
ejpam-5048	304	20	.	.	PUNCT
ejpam-5048	305	1	y.	y.	PROPN
ejpam-5048	305	2	b.	b.	PROPN
ejpam-5048	305	3	jun	jun	PROPN
ejpam-5048	305	4	,	,	PUNCT
ejpam-5048	305	5	ravikumar	ravikumar	PROPN
ejpam-5048	305	6	bandaru	bandaru	PROPN
ejpam-5048	305	7	,	,	PUNCT
ejpam-5048	305	8	rahul	rahul	PROPN
ejpam-5048	305	9	shukla	shukla	PROPN
ejpam-5048	305	10	/	/	SYM
ejpam-5048	305	11	eur	eur	PROPN
ejpam-5048	305	12	.	.	PUNCT
ejpam-5048	306	1	j.	j.	PROPN
ejpam-5048	306	2	pure	pure	PROPN
ejpam-5048	306	3	appl	appl	PROPN
ejpam-5048	306	4	.	.	PROPN
ejpam-5048	306	5	math	math	PROPN
ejpam-5048	306	6	,	,	PUNCT
ejpam-5048	306	7	17	17	NUM
ejpam-5048	306	8	(	(	PUNCT
ejpam-5048	306	9	1	1	NUM
ejpam-5048	306	10	)	)	PUNCT
ejpam-5048	306	11	(	(	PUNCT
ejpam-5048	306	12	2024	2024	NUM
ejpam-5048	306	13	)	)	PUNCT
ejpam-5048	306	14	,	,	PUNCT
ejpam-5048	306	15	569	569	NUM
ejpam-5048	306	16	-	-	SYM
ejpam-5048	306	17	581	581	NUM
ejpam-5048	306	18	579	579	NUM
ejpam-5048	306	19	example	example	NOUN
ejpam-5048	306	20	20	20	NUM
ejpam-5048	306	21	.	.	PUNCT
ejpam-5048	307	1	consider	consider	VERB
ejpam-5048	307	2	the	the	DET
ejpam-5048	307	3	qge	qge	NOUN
ejpam-5048	307	4	-	-	NOUN
ejpam-5048	307	5	algebra	algebra	NOUN
ejpam-5048	307	6	(	(	PUNCT
ejpam-5048	307	7	x	x	X
ejpam-5048	307	8	,	,	PUNCT
ejpam-5048	307	9	∗	∗	NOUN
ejpam-5048	307	10	,	,	PUNCT
ejpam-5048	307	11	1	1	NUM
ejpam-5048	307	12	)	)	PUNCT
ejpam-5048	307	13	given	give	VERB
ejpam-5048	307	14	in	in	ADP
ejpam-5048	307	15	example	example	NOUN
ejpam-5048	307	16	13	13	NUM
ejpam-5048	307	17	.	.	PUNCT
ejpam-5048	308	1	it	it	PRON
ejpam-5048	308	2	is	be	AUX
ejpam-5048	308	3	routine	routine	ADJ
ejpam-5048	308	4	to	to	PART
ejpam-5048	308	5	verify	verify	VERB
ejpam-5048	308	6	that	that	SCONJ
ejpam-5048	308	7	the	the	DET
ejpam-5048	308	8	set	set	NOUN
ejpam-5048	308	9	f	f	NOUN
ejpam-5048	308	10	:	:	PUNCT
ejpam-5048	308	11	=	=	SYM
ejpam-5048	308	12	{	{	PUNCT
ejpam-5048	308	13	1	1	NUM
ejpam-5048	308	14	,	,	PUNCT
ejpam-5048	308	15	e	e	NOUN
ejpam-5048	308	16	}	}	PUNCT
ejpam-5048	308	17	is	be	AUX
ejpam-5048	308	18	a	a	DET
ejpam-5048	308	19	qge	qge	NOUN
ejpam-5048	308	20	-	-	NOUN
ejpam-5048	308	21	filter	filter	NOUN
ejpam-5048	308	22	of	of	ADP
ejpam-5048	308	23	x.	x.	NOUN
ejpam-5048	309	1	but	but	CCONJ
ejpam-5048	309	2	f	f	PROPN
ejpam-5048	309	3	is	be	AUX
ejpam-5048	309	4	not	not	PART
ejpam-5048	309	5	a	a	DET
ejpam-5048	309	6	strong	strong	ADJ
ejpam-5048	309	7	qge	qge	NOUN
ejpam-5048	309	8	-	-	NOUN
ejpam-5048	309	9	filter	filter	NOUN
ejpam-5048	309	10	of	of	ADP
ejpam-5048	309	11	x	x	PRON
ejpam-5048	309	12	since	since	SCONJ
ejpam-5048	309	13	a	a	DET
ejpam-5048	309	14	∗	∗	NOUN
ejpam-5048	309	15	(	(	PUNCT
ejpam-5048	309	16	e	e	NOUN
ejpam-5048	309	17	∗	∗	X
ejpam-5048	309	18	b	b	NOUN
ejpam-5048	309	19	)	)	PUNCT
ejpam-5048	309	20	=	=	PUNCT
ejpam-5048	309	21	a	a	DET
ejpam-5048	309	22	∗	∗	NOUN
ejpam-5048	309	23	d	d	NOUN
ejpam-5048	309	24	=	=	SYM
ejpam-5048	309	25	e	e	PROPN
ejpam-5048	309	26	∈	∈	PROPN
ejpam-5048	309	27	f	f	PROPN
ejpam-5048	309	28	and	and	CCONJ
ejpam-5048	309	29	e	e	PROPN
ejpam-5048	309	30	∈	∈	PROPN
ejpam-5048	309	31	f	f	PROPN
ejpam-5048	309	32	but	but	CCONJ
ejpam-5048	309	33	a	a	DET
ejpam-5048	309	34	∗	∗	NOUN
ejpam-5048	309	35	b	b	NOUN
ejpam-5048	309	36	=	=	SYM
ejpam-5048	309	37	c	c	PROPN
ejpam-5048	309	38	/∈	/∈	PUNCT
ejpam-5048	310	1	f.	f.	PROPN
ejpam-5048	311	1	the	the	DET
ejpam-5048	311	2	following	follow	VERB
ejpam-5048	311	3	example	example	NOUN
ejpam-5048	311	4	shows	show	VERB
ejpam-5048	311	5	that	that	SCONJ
ejpam-5048	311	6	a	a	DET
ejpam-5048	311	7	strong	strong	ADJ
ejpam-5048	311	8	qge	qge	NOUN
ejpam-5048	311	9	-	-	NOUN
ejpam-5048	311	10	filter	filter	NOUN
ejpam-5048	311	11	may	may	AUX
ejpam-5048	311	12	not	not	PART
ejpam-5048	311	13	be	be	AUX
ejpam-5048	311	14	a	a	DET
ejpam-5048	311	15	qge	qge	NOUN
ejpam-5048	311	16	-	-	PUNCT
ejpam-5048	311	17	subalgebra	subalgebra	NOUN
ejpam-5048	311	18	.	.	PUNCT
ejpam-5048	311	19	example	example	NOUN
ejpam-5048	311	20	21	21	NUM
ejpam-5048	311	21	.	.	PUNCT
ejpam-5048	312	1	consider	consider	VERB
ejpam-5048	312	2	the	the	DET
ejpam-5048	312	3	qge	qge	NOUN
ejpam-5048	312	4	-	-	NOUN
ejpam-5048	312	5	algebra	algebra	NOUN
ejpam-5048	312	6	(	(	PUNCT
ejpam-5048	312	7	r	r	NOUN
ejpam-5048	312	8	\	\	PUNCT
ejpam-5048	312	9	{	{	PUNCT
ejpam-5048	312	10	0	0	NUM
ejpam-5048	312	11	}	}	PUNCT
ejpam-5048	312	12	,	,	PUNCT
ejpam-5048	312	13	∗+	∗+	PROPN
ejpam-5048	312	14	,	,	PUNCT
ejpam-5048	312	15	1	1	NUM
ejpam-5048	312	16	)	)	PUNCT
ejpam-5048	312	17	given	give	VERB
ejpam-5048	312	18	in	in	ADP
ejpam-5048	312	19	example	example	NOUN
ejpam-5048	312	20	10	10	NUM
ejpam-5048	312	21	.	.	PUNCT
ejpam-5048	313	1	if	if	SCONJ
ejpam-5048	313	2	we	we	PRON
ejpam-5048	313	3	take	take	VERB
ejpam-5048	313	4	f+	f+	NOUN
ejpam-5048	313	5	:	:	PUNCT
ejpam-5048	313	6	=	=	X
ejpam-5048	313	7	{	{	PUNCT
ejpam-5048	313	8	κ	κ	NOUN
ejpam-5048	313	9	∈	∈	PROPN
ejpam-5048	314	1	r	r	NOUN
ejpam-5048	314	2	|	|	ADV
ejpam-5048	314	3	κ	κ	X
ejpam-5048	314	4	≥	≥	NOUN
ejpam-5048	314	5	1	1	NUM
ejpam-5048	314	6	}	}	PUNCT
ejpam-5048	314	7	,	,	PUNCT
ejpam-5048	314	8	then	then	ADV
ejpam-5048	314	9	1	1	NUM
ejpam-5048	314	10	∈	∈	NOUN
ejpam-5048	314	11	f+	f+	NOUN
ejpam-5048	314	12	⊆	⊆	NUM
ejpam-5048	314	13	r	r	NOUN
ejpam-5048	314	14	\	\	PUNCT
ejpam-5048	314	15	{	{	PUNCT
ejpam-5048	314	16	0	0	NUM
ejpam-5048	314	17	}	}	PUNCT
ejpam-5048	314	18	.	.	PUNCT
ejpam-5048	315	1	let	let	VERB
ejpam-5048	315	2	κ	κ	NOUN
ejpam-5048	315	3	,	,	PUNCT
ejpam-5048	315	4	δ	δ	PROPN
ejpam-5048	315	5	,	,	PUNCT
ejpam-5048	315	6	ς	ς	PROPN
ejpam-5048	315	7	∈	∈	PROPN
ejpam-5048	315	8	r	r	NOUN
ejpam-5048	315	9	\	\	PUNCT
ejpam-5048	315	10	{	{	PUNCT
ejpam-5048	315	11	0	0	NUM
ejpam-5048	315	12	}	}	PUNCT
ejpam-5048	315	13	be	be	AUX
ejpam-5048	315	14	such	such	ADJ
ejpam-5048	315	15	that	that	SCONJ
ejpam-5048	315	16	κ	κ	PROPN
ejpam-5048	315	17	∗	∗	NOUN
ejpam-5048	315	18	(	(	PUNCT
ejpam-5048	315	19	δ	δ	PROPN
ejpam-5048	315	20	∗	∗	PROPN
ejpam-5048	315	21	ς	ς	NOUN
ejpam-5048	315	22	)	)	PUNCT
ejpam-5048	315	23	∈	∈	PROPN
ejpam-5048	315	24	f+	f+	NOUN
ejpam-5048	315	25	and	and	CCONJ
ejpam-5048	315	26	δ	δ	PROPN
ejpam-5048	315	27	∈	∈	PROPN
ejpam-5048	315	28	f+	f+	NOUN
ejpam-5048	315	29	.	.	PUNCT
ejpam-5048	316	1	then	then	ADV
ejpam-5048	316	2	ς	ς	PROPN
ejpam-5048	316	3	κδ	κδ	PROPN
ejpam-5048	316	4	=	=	PROPN
ejpam-5048	316	5	κ	κ	PROPN
ejpam-5048	316	6	∗	∗	NOUN
ejpam-5048	316	7	(	(	PUNCT
ejpam-5048	316	8	δ	δ	PROPN
ejpam-5048	316	9	∗	∗	PROPN
ejpam-5048	316	10	ς	ς	PROPN
ejpam-5048	316	11	)	)	PUNCT
ejpam-5048	316	12	≥	≥	NOUN
ejpam-5048	316	13	1	1	NUM
ejpam-5048	316	14	and	and	CCONJ
ejpam-5048	316	15	δ	δ	PROPN
ejpam-5048	316	16	≥	≥	NUM
ejpam-5048	316	17	1	1	NUM
ejpam-5048	316	18	.	.	PUNCT
ejpam-5048	317	1	it	it	PRON
ejpam-5048	317	2	follows	follow	VERB
ejpam-5048	317	3	that	that	PRON
ejpam-5048	317	4	κ	κ	PROPN
ejpam-5048	317	5	∗	∗	NOUN
ejpam-5048	317	6	ς	ς	PROPN
ejpam-5048	318	1	=	=	SYM
ejpam-5048	318	2	ς	ς	PROPN
ejpam-5048	318	3	κ	κ	PROPN
ejpam-5048	318	4	=	=	PUNCT
ejpam-5048	318	5	δς	δς	NOUN
ejpam-5048	318	6	κδ	κδ	ADP
ejpam-5048	318	7	≥	≥	NOUN
ejpam-5048	318	8	1	1	NUM
ejpam-5048	318	9	,	,	PUNCT
ejpam-5048	318	10	i.e.	i.e.	X
ejpam-5048	318	11	,	,	PUNCT
ejpam-5048	318	12	κ	κ	NOUN
ejpam-5048	318	13	∗	∗	NOUN
ejpam-5048	318	14	ς	ς	PROPN
ejpam-5048	318	15	∈	∈	PROPN
ejpam-5048	318	16	f+	f+	NOUN
ejpam-5048	318	17	.	.	PUNCT
ejpam-5048	319	1	hence	hence	ADV
ejpam-5048	319	2	f+	f+	PROPN
ejpam-5048	319	3	is	be	AUX
ejpam-5048	319	4	a	a	DET
ejpam-5048	319	5	strong	strong	ADJ
ejpam-5048	319	6	qge	qge	NOUN
ejpam-5048	319	7	-	-	NOUN
ejpam-5048	319	8	filter	filter	NOUN
ejpam-5048	319	9	of	of	ADP
ejpam-5048	319	10	r	r	NOUN
ejpam-5048	319	11	\	\	PUNCT
ejpam-5048	319	12	{	{	PUNCT
ejpam-5048	319	13	0	0	NUM
ejpam-5048	319	14	}	}	PUNCT
ejpam-5048	319	15	.	.	PUNCT
ejpam-5048	320	1	but	but	CCONJ
ejpam-5048	320	2	f+	f+	PROPN
ejpam-5048	320	3	is	be	AUX
ejpam-5048	320	4	not	not	PART
ejpam-5048	320	5	a	a	DET
ejpam-5048	320	6	qge	qge	NOUN
ejpam-5048	320	7	-	-	PUNCT
ejpam-5048	320	8	subalgebra	subalgebra	NOUN
ejpam-5048	320	9	of	of	ADP
ejpam-5048	320	10	r	r	NOUN
ejpam-5048	320	11	\	\	PUNCT
ejpam-5048	320	12	{	{	PUNCT
ejpam-5048	320	13	0	0	NUM
ejpam-5048	320	14	}	}	PUNCT
ejpam-5048	320	15	because	because	SCONJ
ejpam-5048	320	16	of	of	ADP
ejpam-5048	320	17	3.5	3.5	NUM
ejpam-5048	320	18	∗	∗	NOUN
ejpam-5048	320	19	2.5	2.5	NUM
ejpam-5048	320	20	=	=	SYM
ejpam-5048	320	21	2.5	2.5	NUM
ejpam-5048	320	22	3.5	3.5	NUM
ejpam-5048	320	23	<	<	SYM
ejpam-5048	320	24	1	1	NUM
ejpam-5048	321	1	and	and	CCONJ
ejpam-5048	321	2	so	so	ADV
ejpam-5048	321	3	3.5	3.5	NUM
ejpam-5048	321	4	∗	∗	NOUN
ejpam-5048	321	5	2.5	2.5	NUM
ejpam-5048	321	6	/∈	/∈	PUNCT
ejpam-5048	321	7	f+	f+	PROPN
ejpam-5048	321	8	for	for	ADP
ejpam-5048	321	9	2.5	2.5	NUM
ejpam-5048	321	10	,	,	PUNCT
ejpam-5048	321	11	3.5	3.5	NUM
ejpam-5048	321	12	∈	∈	PROPN
ejpam-5048	321	13	f+	f+	NOUN
ejpam-5048	321	14	.	.	PUNCT
ejpam-5048	322	1	in	in	ADP
ejpam-5048	322	2	example	example	NOUN
ejpam-5048	322	3	10	10	NUM
ejpam-5048	322	4	,	,	PUNCT
ejpam-5048	322	5	the	the	DET
ejpam-5048	322	6	set	set	ADJ
ejpam-5048	322	7	f−	f−	NOUN
ejpam-5048	322	8	:	:	PUNCT
ejpam-5048	322	9	=	=	SYM
ejpam-5048	322	10	{	{	PUNCT
ejpam-5048	322	11	κ	κ	NOUN
ejpam-5048	322	12	∈	∈	PROPN
ejpam-5048	322	13	r	r	NOUN
ejpam-5048	323	1	|	|	ADV
ejpam-5048	323	2	κ	κ	X
ejpam-5048	323	3	≤	≤	NUM
ejpam-5048	323	4	−1	−1	NOUN
ejpam-5048	323	5	}	}	PUNCT
ejpam-5048	323	6	is	be	AUX
ejpam-5048	323	7	neither	neither	CCONJ
ejpam-5048	323	8	a	a	DET
ejpam-5048	323	9	strong	strong	ADJ
ejpam-5048	323	10	qge	qge	NOUN
ejpam-5048	323	11	-	-	NOUN
ejpam-5048	323	12	filter	filter	NOUN
ejpam-5048	323	13	nor	nor	CCONJ
ejpam-5048	323	14	a	a	DET
ejpam-5048	323	15	qge	qge	NOUN
ejpam-5048	323	16	-	-	NOUN
ejpam-5048	323	17	subalgebra	subalgebra	NOUN
ejpam-5048	323	18	,	,	PUNCT
ejpam-5048	323	19	as	as	SCONJ
ejpam-5048	323	20	checked	check	VERB
ejpam-5048	323	21	in	in	ADP
ejpam-5048	323	22	the	the	DET
ejpam-5048	323	23	following	follow	VERB
ejpam-5048	323	24	example	example	NOUN
ejpam-5048	323	25	.	.	PUNCT
ejpam-5048	324	1	example	example	NOUN
ejpam-5048	324	2	22	22	NUM
ejpam-5048	324	3	.	.	PUNCT
ejpam-5048	325	1	consider	consider	VERB
ejpam-5048	325	2	the	the	DET
ejpam-5048	325	3	qge	qge	NOUN
ejpam-5048	325	4	-	-	NOUN
ejpam-5048	325	5	algebra	algebra	NOUN
ejpam-5048	325	6	(	(	PUNCT
ejpam-5048	325	7	r	r	NOUN
ejpam-5048	325	8	\	\	PUNCT
ejpam-5048	325	9	{	{	PUNCT
ejpam-5048	325	10	0	0	NUM
ejpam-5048	325	11	}	}	PUNCT
ejpam-5048	325	12	,	,	PUNCT
ejpam-5048	325	13	∗+	∗+	PROPN
ejpam-5048	325	14	,	,	PUNCT
ejpam-5048	325	15	1	1	NUM
ejpam-5048	325	16	)	)	PUNCT
ejpam-5048	325	17	given	give	VERB
ejpam-5048	325	18	in	in	ADP
ejpam-5048	325	19	example	example	NOUN
ejpam-5048	325	20	10	10	NUM
ejpam-5048	325	21	.	.	PUNCT
ejpam-5048	326	1	let	let	VERB
ejpam-5048	326	2	f−	f−	NOUN
ejpam-5048	326	3	:	:	PUNCT
ejpam-5048	326	4	=	=	X
ejpam-5048	326	5	{	{	PUNCT
ejpam-5048	326	6	κ	κ	NOUN
ejpam-5048	326	7	∈	∈	PROPN
ejpam-5048	327	1	r	r	NOUN
ejpam-5048	327	2	|	|	NOUN
ejpam-5048	327	3	κ	κ	X
ejpam-5048	327	4	≤	≤	NUM
ejpam-5048	327	5	−1	−1	NOUN
ejpam-5048	327	6	}	}	PUNCT
ejpam-5048	327	7	.	.	PUNCT
ejpam-5048	328	1	then	then	ADV
ejpam-5048	328	2	−1	−1	NOUN
ejpam-5048	328	3	∈	∈	NOUN
ejpam-5048	328	4	f−	f−	PROPN
ejpam-5048	328	5	⊆	⊆	NUM
ejpam-5048	328	6	r\{0	r\{0	NOUN
ejpam-5048	328	7	}	}	PUNCT
ejpam-5048	328	8	.	.	PUNCT
ejpam-5048	329	1	let	let	VERB
ejpam-5048	329	2	κ	κ	NOUN
ejpam-5048	329	3	,	,	PUNCT
ejpam-5048	329	4	δ	δ	PROPN
ejpam-5048	329	5	,	,	PUNCT
ejpam-5048	329	6	ς	ς	PROPN
ejpam-5048	329	7	∈	∈	PROPN
ejpam-5048	329	8	r\{0	r\{0	PROPN
ejpam-5048	329	9	}	}	PUNCT
ejpam-5048	329	10	be	be	AUX
ejpam-5048	329	11	such	such	ADJ
ejpam-5048	329	12	that	that	DET
ejpam-5048	329	13	κ∗(δ∗ς	κ∗(δ∗ς	NOUN
ejpam-5048	329	14	)	)	PUNCT
ejpam-5048	329	15	∈	∈	PROPN
ejpam-5048	329	16	f−	f−	PROPN
ejpam-5048	329	17	and	and	CCONJ
ejpam-5048	329	18	δ	δ	PROPN
ejpam-5048	329	19	∈	∈	PROPN
ejpam-5048	329	20	f−.	f−.	NOUN
ejpam-5048	329	21	then	then	ADV
ejpam-5048	329	22	ς	ς	PROPN
ejpam-5048	329	23	κδ	κδ	NOUN
ejpam-5048	329	24	=	=	PROPN
ejpam-5048	329	25	κ	κ	PROPN
ejpam-5048	329	26	∗	∗	NOUN
ejpam-5048	329	27	(	(	PUNCT
ejpam-5048	329	28	δ	δ	PROPN
ejpam-5048	329	29	∗	∗	PROPN
ejpam-5048	329	30	ς	ς	NOUN
ejpam-5048	329	31	)	)	PUNCT
ejpam-5048	329	32	≤	≤	NOUN
ejpam-5048	329	33	−1	−1	NOUN
ejpam-5048	329	34	and	and	CCONJ
ejpam-5048	329	35	δ	δ	PROPN
ejpam-5048	329	36	≤	≤	NOUN
ejpam-5048	329	37	−1	−1	NOUN
ejpam-5048	329	38	.	.	PUNCT
ejpam-5048	330	1	but	but	CCONJ
ejpam-5048	330	2	κ	κ	NOUN
ejpam-5048	330	3	∗	∗	NOUN
ejpam-5048	330	4	ς	ς	PROPN
ejpam-5048	331	1	=	=	SYM
ejpam-5048	331	2	ς	ς	PROPN
ejpam-5048	331	3	κ	κ	PROPN
ejpam-5048	331	4	=	=	PUNCT
ejpam-5048	331	5	δς	δς	NOUN
ejpam-5048	331	6	κδ	κδ	ADP
ejpam-5048	331	7	≥	≥	NOUN
ejpam-5048	331	8	0	0	NUM
ejpam-5048	331	9	,	,	PUNCT
ejpam-5048	331	10	i.e.	i.e.	X
ejpam-5048	331	11	,	,	PUNCT
ejpam-5048	331	12	κ	κ	NOUN
ejpam-5048	331	13	∗	∗	NOUN
ejpam-5048	331	14	ς	ς	PROPN
ejpam-5048	331	15	/∈	/∈	PUNCT
ejpam-5048	331	16	f−.	f−.	VERB
ejpam-5048	331	17	thus	thus	ADV
ejpam-5048	331	18	f−	f−	PROPN
ejpam-5048	331	19	is	be	AUX
ejpam-5048	331	20	not	not	PART
ejpam-5048	331	21	a	a	DET
ejpam-5048	331	22	strong	strong	ADJ
ejpam-5048	331	23	qge	qge	NOUN
ejpam-5048	331	24	-	-	NOUN
ejpam-5048	331	25	filter	filter	NOUN
ejpam-5048	331	26	of	of	ADP
ejpam-5048	331	27	r	r	NOUN
ejpam-5048	331	28	\	\	PUNCT
ejpam-5048	331	29	{	{	PUNCT
ejpam-5048	331	30	0	0	NUM
ejpam-5048	331	31	}	}	PUNCT
ejpam-5048	331	32	.	.	PUNCT
ejpam-5048	332	1	also	also	ADV
ejpam-5048	332	2	if	if	SCONJ
ejpam-5048	332	3	κ	κ	X
ejpam-5048	332	4	,	,	PUNCT
ejpam-5048	332	5	δ	δ	PROPN
ejpam-5048	332	6	∈	∈	PROPN
ejpam-5048	332	7	f−	f−	PROPN
ejpam-5048	332	8	,	,	PUNCT
ejpam-5048	332	9	then	then	ADV
ejpam-5048	332	10	κ	κ	X
ejpam-5048	332	11	≤	≤	NUM
ejpam-5048	332	12	−1	−1	NOUN
ejpam-5048	332	13	nd	nd	PART
ejpam-5048	332	14	δ	δ	NOUN
ejpam-5048	332	15	≤	≤	NOUN
ejpam-5048	332	16	−1	−1	NOUN
ejpam-5048	332	17	.	.	PUNCT
ejpam-5048	333	1	hence	hence	ADV
ejpam-5048	333	2	κ	κ	ADP
ejpam-5048	333	3	∗	∗	X
ejpam-5048	333	4	δ	δ	X
ejpam-5048	334	1	=	=	PUNCT
ejpam-5048	334	2	δ	δ	PROPN
ejpam-5048	334	3	κ	κ	X
ejpam-5048	334	4	≥	≥	NOUN
ejpam-5048	334	5	0	0	NUM
ejpam-5048	334	6	,	,	PUNCT
ejpam-5048	334	7	that	that	ADV
ejpam-5048	334	8	is	is	ADV
ejpam-5048	334	9	,	,	PUNCT
ejpam-5048	334	10	κ	κ	PROPN
ejpam-5048	334	11	∗	∗	NOUN
ejpam-5048	334	12	δ	δ	PROPN
ejpam-5048	334	13	/∈	/∈	PUNCT
ejpam-5048	334	14	f−.	f−.	VERB
ejpam-5048	334	15	therefore	therefore	ADV
ejpam-5048	334	16	f−	f−	PROPN
ejpam-5048	334	17	is	be	AUX
ejpam-5048	334	18	not	not	PART
ejpam-5048	334	19	a	a	DET
ejpam-5048	334	20	strong	strong	ADJ
ejpam-5048	334	21	qge	qge	NOUN
ejpam-5048	334	22	-	-	PUNCT
ejpam-5048	334	23	subalgebra	subalgebra	NOUN
ejpam-5048	334	24	of	of	ADP
ejpam-5048	334	25	r	r	NOUN
ejpam-5048	334	26	\	\	PUNCT
ejpam-5048	334	27	{	{	PUNCT
ejpam-5048	334	28	0	0	NUM
ejpam-5048	334	29	}	}	PUNCT
ejpam-5048	334	30	.	.	PUNCT
ejpam-5048	335	1	the	the	DET
ejpam-5048	335	2	following	follow	VERB
ejpam-5048	335	3	example	example	NOUN
ejpam-5048	335	4	shows	show	VERB
ejpam-5048	335	5	that	that	SCONJ
ejpam-5048	335	6	a	a	DET
ejpam-5048	335	7	qge	qge	NOUN
ejpam-5048	335	8	-	-	PUNCT
ejpam-5048	335	9	subalgebra	subalgebra	NOUN
ejpam-5048	335	10	may	may	AUX
ejpam-5048	335	11	not	not	PART
ejpam-5048	335	12	be	be	AUX
ejpam-5048	335	13	a	a	DET
ejpam-5048	335	14	strong	strong	ADJ
ejpam-5048	335	15	qge	qge	NOUN
ejpam-5048	335	16	-	-	NOUN
ejpam-5048	335	17	filter	filter	NOUN
ejpam-5048	335	18	.	.	PUNCT
ejpam-5048	335	19	example	example	NOUN
ejpam-5048	335	20	23	23	NUM
ejpam-5048	335	21	.	.	PUNCT
ejpam-5048	336	1	consider	consider	VERB
ejpam-5048	336	2	the	the	DET
ejpam-5048	336	3	qge	qge	NOUN
ejpam-5048	336	4	-	-	NOUN
ejpam-5048	336	5	algebra	algebra	NOUN
ejpam-5048	336	6	(	(	PUNCT
ejpam-5048	336	7	x	x	X
ejpam-5048	336	8	,	,	PUNCT
ejpam-5048	336	9	∗	∗	NOUN
ejpam-5048	336	10	,	,	PUNCT
ejpam-5048	336	11	1	1	NUM
ejpam-5048	336	12	)	)	PUNCT
ejpam-5048	336	13	given	give	VERB
ejpam-5048	336	14	in	in	ADP
ejpam-5048	336	15	example	example	NOUN
ejpam-5048	336	16	13	13	NUM
ejpam-5048	336	17	.	.	PUNCT
ejpam-5048	337	1	it	it	PRON
ejpam-5048	337	2	is	be	AUX
ejpam-5048	337	3	routine	routine	ADJ
ejpam-5048	337	4	to	to	PART
ejpam-5048	337	5	verify	verify	VERB
ejpam-5048	337	6	that	that	SCONJ
ejpam-5048	337	7	the	the	DET
ejpam-5048	337	8	set	set	NOUN
ejpam-5048	337	9	f	f	NOUN
ejpam-5048	337	10	:	:	PUNCT
ejpam-5048	337	11	=	=	SYM
ejpam-5048	337	12	{	{	PUNCT
ejpam-5048	337	13	1	1	NUM
ejpam-5048	337	14	,	,	PUNCT
ejpam-5048	337	15	e	e	NOUN
ejpam-5048	337	16	}	}	PUNCT
ejpam-5048	337	17	is	be	AUX
ejpam-5048	337	18	a	a	DET
ejpam-5048	337	19	qge	qge	NOUN
ejpam-5048	337	20	-	-	PUNCT
ejpam-5048	337	21	subalgebra	subalgebra	NOUN
ejpam-5048	337	22	of	of	ADP
ejpam-5048	337	23	x.	x.	NOUN
ejpam-5048	338	1	but	but	CCONJ
ejpam-5048	338	2	f	f	PROPN
ejpam-5048	338	3	is	be	AUX
ejpam-5048	338	4	not	not	PART
ejpam-5048	338	5	a	a	DET
ejpam-5048	338	6	strong	strong	ADJ
ejpam-5048	338	7	qge	qge	NOUN
ejpam-5048	338	8	-	-	NOUN
ejpam-5048	338	9	filter	filter	NOUN
ejpam-5048	338	10	of	of	ADP
ejpam-5048	338	11	x	x	PRON
ejpam-5048	338	12	since	since	SCONJ
ejpam-5048	338	13	a	a	DET
ejpam-5048	338	14	∗	∗	NOUN
ejpam-5048	338	15	(	(	PUNCT
ejpam-5048	338	16	e	e	NOUN
ejpam-5048	338	17	∗	∗	X
ejpam-5048	338	18	b	b	NOUN
ejpam-5048	338	19	)	)	PUNCT
ejpam-5048	338	20	=	=	PUNCT
ejpam-5048	338	21	a	a	DET
ejpam-5048	338	22	∗	∗	NOUN
ejpam-5048	338	23	d	d	NOUN
ejpam-5048	338	24	=	=	SYM
ejpam-5048	338	25	e	e	PROPN
ejpam-5048	338	26	∈	∈	PROPN
ejpam-5048	338	27	f	f	PROPN
ejpam-5048	338	28	and	and	CCONJ
ejpam-5048	338	29	e	e	PROPN
ejpam-5048	338	30	∈	∈	PROPN
ejpam-5048	338	31	f	f	PROPN
ejpam-5048	338	32	but	but	CCONJ
ejpam-5048	338	33	a	a	DET
ejpam-5048	338	34	∗	∗	NOUN
ejpam-5048	338	35	b	b	NOUN
ejpam-5048	338	36	=	=	SYM
ejpam-5048	338	37	c	c	PROPN
ejpam-5048	338	38	/∈	/∈	PUNCT
ejpam-5048	339	1	f.	f.	PROPN
ejpam-5048	339	2	by	by	ADP
ejpam-5048	339	3	examples	example	NOUN
ejpam-5048	339	4	21	21	NUM
ejpam-5048	339	5	and	and	CCONJ
ejpam-5048	339	6	23	23	NUM
ejpam-5048	339	7	,	,	PUNCT
ejpam-5048	339	8	we	we	PRON
ejpam-5048	339	9	can	can	AUX
ejpam-5048	339	10	see	see	VERB
ejpam-5048	339	11	that	that	SCONJ
ejpam-5048	339	12	the	the	DET
ejpam-5048	339	13	two	two	NUM
ejpam-5048	339	14	concepts	concept	NOUN
ejpam-5048	339	15	qge	qge	NOUN
ejpam-5048	339	16	-	-	PUNCT
ejpam-5048	339	17	subalgebra	subalgebra	ADJ
ejpam-5048	339	18	and	and	CCONJ
ejpam-5048	339	19	strong	strong	ADJ
ejpam-5048	339	20	qge	qge	NOUN
ejpam-5048	339	21	-	-	NOUN
ejpam-5048	339	22	filter	filter	NOUN
ejpam-5048	339	23	are	be	AUX
ejpam-5048	339	24	independent	independent	ADJ
ejpam-5048	339	25	of	of	ADP
ejpam-5048	339	26	each	each	DET
ejpam-5048	339	27	other	other	ADJ
ejpam-5048	339	28	.	.	PUNCT
ejpam-5048	340	1	we	we	PRON
ejpam-5048	340	2	discuss	discuss	VERB
ejpam-5048	340	3	relationship	relationship	NOUN
ejpam-5048	340	4	between	between	ADP
ejpam-5048	340	5	a	a	DET
ejpam-5048	340	6	qge	qge	NOUN
ejpam-5048	340	7	-	-	PUNCT
ejpam-5048	340	8	subalgebra	subalgebra	NOUN
ejpam-5048	340	9	and	and	CCONJ
ejpam-5048	340	10	a	a	DET
ejpam-5048	340	11	qge	qge	NOUN
ejpam-5048	340	12	-	-	NOUN
ejpam-5048	340	13	filter	filter	NOUN
ejpam-5048	340	14	.	.	PUNCT
ejpam-5048	341	1	theorem	theorem	ADJ
ejpam-5048	341	2	10	10	NUM
ejpam-5048	341	3	.	.	PUNCT
ejpam-5048	342	1	every	every	DET
ejpam-5048	342	2	qge	qge	NOUN
ejpam-5048	342	3	-	-	PUNCT
ejpam-5048	342	4	subalgebra	subalgebra	NOUN
ejpam-5048	342	5	is	be	AUX
ejpam-5048	342	6	a	a	DET
ejpam-5048	342	7	qge	qge	NOUN
ejpam-5048	342	8	-	-	NOUN
ejpam-5048	342	9	filter	filter	NOUN
ejpam-5048	342	10	.	.	PUNCT
ejpam-5048	343	1	proof	proof	NOUN
ejpam-5048	343	2	.	.	PUNCT
ejpam-5048	344	1	let	let	AUX
ejpam-5048	344	2	e	e	PRON
ejpam-5048	344	3	be	be	AUX
ejpam-5048	344	4	a	a	DET
ejpam-5048	344	5	qge	qge	NOUN
ejpam-5048	344	6	-	-	PUNCT
ejpam-5048	344	7	subalgebra	subalgebra	NOUN
ejpam-5048	344	8	of	of	ADP
ejpam-5048	344	9	x.	x.	NOUN
ejpam-5048	344	10	proposition	proposition	PROPN
ejpam-5048	344	11	3	3	NUM
ejpam-5048	344	12	shows	show	VERB
ejpam-5048	344	13	that	that	SCONJ
ejpam-5048	344	14	1	1	NUM
ejpam-5048	344	15	∈	∈	PROPN
ejpam-5048	344	16	e.	e.	NOUN
ejpam-5048	344	17	let	let	VERB
ejpam-5048	344	18	κ	κ	VERB
ejpam-5048	344	19	,	,	PUNCT
ejpam-5048	344	20	δ	δ	PROPN
ejpam-5048	344	21	∈	∈	PROPN
ejpam-5048	344	22	x	x	AUX
ejpam-5048	344	23	be	be	AUX
ejpam-5048	344	24	such	such	ADJ
ejpam-5048	344	25	that	that	SCONJ
ejpam-5048	344	26	κ∗δ	κ∗δ	PUNCT
ejpam-5048	344	27	∈	∈	PROPN
ejpam-5048	344	28	e	e	NOUN
ejpam-5048	344	29	and	and	CCONJ
ejpam-5048	344	30	κ	κ	PROPN
ejpam-5048	344	31	∈	∈	PROPN
ejpam-5048	344	32	e.	e.	PROPN
ejpam-5048	344	33	then	then	ADV
ejpam-5048	344	34	κ∗1	κ∗1	PROPN
ejpam-5048	344	35	∈	∈	PROPN
ejpam-5048	344	36	e	e	X
ejpam-5048	344	37	by	by	ADP
ejpam-5048	344	38	(	(	PUNCT
ejpam-5048	344	39	19	19	NUM
ejpam-5048	344	40	)	)	PUNCT
ejpam-5048	344	41	,	,	PUNCT
ejpam-5048	344	42	and	and	CCONJ
ejpam-5048	344	43	so	so	ADV
ejpam-5048	344	44	δ	δ	PROPN
ejpam-5048	344	45	=	=	SYM
ejpam-5048	344	46	1∗δ	1∗δ	NUM
ejpam-5048	344	47	=	=	SYM
ejpam-5048	344	48	(	(	PUNCT
ejpam-5048	344	49	κ∗1)∗(κ∗δ	κ∗1)∗(κ∗δ	NOUN
ejpam-5048	344	50	)	)	PUNCT
ejpam-5048	344	51	∈	∈	PROPN
ejpam-5048	344	52	e	e	X
ejpam-5048	344	53	by	by	ADP
ejpam-5048	344	54	(	(	PUNCT
ejpam-5048	344	55	ge2	ge2	NOUN
ejpam-5048	344	56	)	)	PUNCT
ejpam-5048	344	57	,	,	PUNCT
ejpam-5048	344	58	(	(	PUNCT
ejpam-5048	344	59	10	10	NUM
ejpam-5048	344	60	)	)	PUNCT
ejpam-5048	344	61	and	and	CCONJ
ejpam-5048	344	62	(	(	PUNCT
ejpam-5048	344	63	19	19	NUM
ejpam-5048	344	64	)	)	PUNCT
ejpam-5048	344	65	.	.	PUNCT
ejpam-5048	345	1	therefore	therefore	ADV
ejpam-5048	345	2	e	e	PROPN
ejpam-5048	345	3	is	be	AUX
ejpam-5048	345	4	a	a	DET
ejpam-5048	345	5	qge	qge	NOUN
ejpam-5048	345	6	-	-	NOUN
ejpam-5048	345	7	filter	filter	NOUN
ejpam-5048	345	8	of	of	ADP
ejpam-5048	345	9	x.	x.	NOUN
ejpam-5048	345	10	in	in	ADP
ejpam-5048	345	11	the	the	DET
ejpam-5048	345	12	following	follow	VERB
ejpam-5048	345	13	example	example	NOUN
ejpam-5048	345	14	,	,	PUNCT
ejpam-5048	345	15	we	we	PRON
ejpam-5048	345	16	know	know	VERB
ejpam-5048	345	17	that	that	SCONJ
ejpam-5048	345	18	the	the	DET
ejpam-5048	345	19	converse	converse	NOUN
ejpam-5048	345	20	of	of	ADP
ejpam-5048	345	21	theorem	theorem	NOUN
ejpam-5048	345	22	10	10	NUM
ejpam-5048	345	23	may	may	AUX
ejpam-5048	345	24	not	not	PART
ejpam-5048	345	25	be	be	AUX
ejpam-5048	345	26	true	true	ADJ
ejpam-5048	345	27	.	.	PUNCT
ejpam-5048	346	1	example	example	NOUN
ejpam-5048	346	2	24	24	NUM
ejpam-5048	346	3	.	.	PUNCT
ejpam-5048	347	1	consider	consider	VERB
ejpam-5048	347	2	the	the	DET
ejpam-5048	347	3	qge	qge	NOUN
ejpam-5048	347	4	-	-	NOUN
ejpam-5048	347	5	filter	filter	NOUN
ejpam-5048	347	6	k	k	NOUN
ejpam-5048	348	1	:	:	PUNCT
ejpam-5048	348	2	=	=	SYM
ejpam-5048	348	3	x	x	SYM
ejpam-5048	348	4	×	×	PROPN
ejpam-5048	348	5	n0	n0	NUM
ejpam-5048	348	6	of	of	ADP
ejpam-5048	348	7	y	y	PRON
ejpam-5048	348	8	which	which	PRON
ejpam-5048	348	9	is	be	AUX
ejpam-5048	348	10	described	describe	VERB
ejpam-5048	348	11	in	in	ADP
ejpam-5048	348	12	example	example	NOUN
ejpam-5048	348	13	14	14	NUM
ejpam-5048	348	14	.	.	PUNCT
ejpam-5048	349	1	since	since	SCONJ
ejpam-5048	349	2	(	(	PUNCT
ejpam-5048	349	3	κ1	κ1	NOUN
ejpam-5048	349	4	,	,	PUNCT
ejpam-5048	349	5	7	7	NUM
ejpam-5048	349	6	)	)	PUNCT
ejpam-5048	349	7	∗	∗	NOUN
ejpam-5048	349	8	(	(	PUNCT
ejpam-5048	349	9	κ2	κ2	NOUN
ejpam-5048	349	10	,	,	PUNCT
ejpam-5048	349	11	3	3	NUM
ejpam-5048	349	12	)	)	PUNCT
ejpam-5048	349	13	=	=	SYM
ejpam-5048	349	14	(	(	PUNCT
ejpam-5048	349	15	κ1	κ1	NOUN
ejpam-5048	349	16	∗x	∗x	PROPN
ejpam-5048	349	17	κ2,−4	κ2,−4	PROPN
ejpam-5048	349	18	)	)	PUNCT
ejpam-5048	349	19	/∈	/∈	PUNCT
ejpam-5048	350	1	k	k	PROPN
ejpam-5048	350	2	for	for	ADP
ejpam-5048	350	3	all	all	DET
ejpam-5048	350	4	κ1	κ1	NOUN
ejpam-5048	350	5	,	,	PUNCT
ejpam-5048	350	6	κ2	κ2	NOUN
ejpam-5048	350	7	∈	∈	PROPN
ejpam-5048	351	1	x	x	X
ejpam-5048	351	2	,	,	PUNCT
ejpam-5048	351	3	we	we	PRON
ejpam-5048	351	4	know	know	VERB
ejpam-5048	351	5	that	that	SCONJ
ejpam-5048	351	6	k	k	PROPN
ejpam-5048	351	7	is	be	AUX
ejpam-5048	351	8	not	not	PART
ejpam-5048	351	9	a	a	DET
ejpam-5048	351	10	qge	qge	NOUN
ejpam-5048	351	11	-	-	PUNCT
ejpam-5048	351	12	subalgebra	subalgebra	NOUN
ejpam-5048	351	13	of	of	ADP
ejpam-5048	351	14	y	y	PROPN
ejpam-5048	351	15	.	.	PUNCT
ejpam-5048	352	1	definition	definition	NOUN
ejpam-5048	352	2	7	7	NUM
ejpam-5048	352	3	.	.	PUNCT
ejpam-5048	353	1	a	a	DET
ejpam-5048	353	2	qge	qge	NOUN
ejpam-5048	353	3	-	-	NOUN
ejpam-5048	353	4	filter	filter	NOUN
ejpam-5048	353	5	f	f	NOUN
ejpam-5048	353	6	of	of	ADP
ejpam-5048	353	7	x	x	PROPN
ejpam-5048	353	8	is	be	AUX
ejpam-5048	353	9	said	say	VERB
ejpam-5048	353	10	to	to	PART
ejpam-5048	353	11	be	be	AUX
ejpam-5048	353	12	closed	close	VERB
ejpam-5048	353	13	if	if	SCONJ
ejpam-5048	353	14	f	f	PROPN
ejpam-5048	353	15	is	be	AUX
ejpam-5048	353	16	closed	close	VERB
ejpam-5048	353	17	under	under	ADP
ejpam-5048	353	18	the	the	DET
ejpam-5048	353	19	binary	binary	ADJ
ejpam-5048	353	20	operation	operation	NOUN
ejpam-5048	353	21	“	"	PUNCT
ejpam-5048	353	22	∗	∗	NOUN
ejpam-5048	353	23	”	"	PUNCT
ejpam-5048	353	24	on	on	ADP
ejpam-5048	353	25	x	x	NOUN
ejpam-5048	353	26	,	,	PUNCT
ejpam-5048	353	27	i.e.	i.e.	X
ejpam-5048	353	28	,	,	PUNCT
ejpam-5048	353	29	f	f	PROPN
ejpam-5048	353	30	is	be	AUX
ejpam-5048	353	31	a	a	DET
ejpam-5048	353	32	qge	qge	NOUN
ejpam-5048	353	33	-	-	PUNCT
ejpam-5048	353	34	subalgebra	subalgebra	NOUN
ejpam-5048	353	35	of	of	ADP
ejpam-5048	353	36	x.	x.	PROPN
ejpam-5048	353	37	y.	y.	PROPN
ejpam-5048	353	38	b.	b.	PROPN
ejpam-5048	353	39	jun	jun	PROPN
ejpam-5048	353	40	,	,	PUNCT
ejpam-5048	353	41	ravikumar	ravikumar	PROPN
ejpam-5048	353	42	bandaru	bandaru	PROPN
ejpam-5048	353	43	,	,	PUNCT
ejpam-5048	353	44	rahul	rahul	PROPN
ejpam-5048	353	45	shukla	shukla	PROPN
ejpam-5048	353	46	/	/	SYM
ejpam-5048	353	47	eur	eur	PROPN
ejpam-5048	353	48	.	.	PUNCT
ejpam-5048	354	1	j.	j.	PROPN
ejpam-5048	354	2	pure	pure	PROPN
ejpam-5048	354	3	appl	appl	PROPN
ejpam-5048	354	4	.	.	PROPN
ejpam-5048	354	5	math	math	PROPN
ejpam-5048	354	6	,	,	PUNCT
ejpam-5048	354	7	17	17	NUM
ejpam-5048	354	8	(	(	PUNCT
ejpam-5048	354	9	1	1	NUM
ejpam-5048	354	10	)	)	PUNCT
ejpam-5048	354	11	(	(	PUNCT
ejpam-5048	354	12	2024	2024	NUM
ejpam-5048	354	13	)	)	PUNCT
ejpam-5048	354	14	,	,	PUNCT
ejpam-5048	354	15	569	569	NUM
ejpam-5048	354	16	-	-	SYM
ejpam-5048	354	17	581	581	NUM
ejpam-5048	354	18	580	580	NUM
ejpam-5048	354	19	example	example	NOUN
ejpam-5048	354	20	25	25	NUM
ejpam-5048	354	21	.	.	PUNCT
ejpam-5048	354	22	consider	consider	VERB
ejpam-5048	354	23	the	the	DET
ejpam-5048	354	24	qge	qge	NOUN
ejpam-5048	354	25	-	-	NOUN
ejpam-5048	354	26	algebra	algebra	NOUN
ejpam-5048	354	27	x	x	PUNCT
ejpam-5048	354	28	given	give	VERB
ejpam-5048	354	29	in	in	ADP
ejpam-5048	354	30	example	example	NOUN
ejpam-5048	354	31	16	16	NUM
ejpam-5048	354	32	.	.	PUNCT
ejpam-5048	355	1	it	it	PRON
ejpam-5048	355	2	is	be	AUX
ejpam-5048	355	3	routine	routine	ADJ
ejpam-5048	355	4	to	to	PART
ejpam-5048	355	5	verify	verify	VERB
ejpam-5048	355	6	that	that	SCONJ
ejpam-5048	355	7	the	the	DET
ejpam-5048	355	8	set	set	NOUN
ejpam-5048	355	9	f	f	X
ejpam-5048	355	10	=	=	PUNCT
ejpam-5048	355	11	{	{	PUNCT
ejpam-5048	355	12	1	1	NUM
ejpam-5048	355	13	,	,	PUNCT
ejpam-5048	355	14	b	b	NOUN
ejpam-5048	355	15	,	,	PUNCT
ejpam-5048	355	16	c	c	NOUN
ejpam-5048	355	17	}	}	PUNCT
ejpam-5048	355	18	is	be	AUX
ejpam-5048	355	19	a	a	DET
ejpam-5048	355	20	closed	closed	ADJ
ejpam-5048	355	21	qge	qge	NOUN
ejpam-5048	355	22	-	-	NOUN
ejpam-5048	355	23	filter	filter	NOUN
ejpam-5048	355	24	of	of	ADP
ejpam-5048	355	25	x.	x.	PROPN
ejpam-5048	355	26	example	example	NOUN
ejpam-5048	355	27	26	26	NUM
ejpam-5048	355	28	.	.	PUNCT
ejpam-5048	356	1	consider	consider	VERB
ejpam-5048	356	2	the	the	DET
ejpam-5048	356	3	qge	qge	NOUN
ejpam-5048	356	4	-	-	NOUN
ejpam-5048	356	5	algebra	algebra	NOUN
ejpam-5048	356	6	(	(	PUNCT
ejpam-5048	356	7	r	r	NOUN
ejpam-5048	356	8	,	,	PUNCT
ejpam-5048	356	9	∗	∗	NOUN
ejpam-5048	356	10	,	,	PUNCT
ejpam-5048	356	11	0	0	NUM
ejpam-5048	356	12	)	)	PUNCT
ejpam-5048	356	13	in	in	ADP
ejpam-5048	356	14	example	example	NOUN
ejpam-5048	356	15	4	4	X
ejpam-5048	356	16	.	.	PUNCT
ejpam-5048	357	1	it	it	PRON
ejpam-5048	357	2	is	be	AUX
ejpam-5048	357	3	routine	routine	ADJ
ejpam-5048	357	4	to	to	PART
ejpam-5048	357	5	verify	verify	VERB
ejpam-5048	357	6	that	that	SCONJ
ejpam-5048	357	7	(	(	PUNCT
ejpam-5048	357	8	z	z	X
ejpam-5048	357	9	,	,	PUNCT
ejpam-5048	357	10	∗	∗	NOUN
ejpam-5048	357	11	,	,	PUNCT
ejpam-5048	357	12	0	0	NUM
ejpam-5048	357	13	)	)	PUNCT
ejpam-5048	357	14	is	be	AUX
ejpam-5048	357	15	a	a	DET
ejpam-5048	357	16	closed	closed	ADJ
ejpam-5048	357	17	qge	qge	NOUN
ejpam-5048	357	18	-	-	NOUN
ejpam-5048	357	19	filter	filter	NOUN
ejpam-5048	357	20	of	of	ADP
ejpam-5048	357	21	(	(	PUNCT
ejpam-5048	357	22	r	r	NOUN
ejpam-5048	357	23	,	,	PUNCT
ejpam-5048	357	24	∗	∗	NOUN
ejpam-5048	357	25	,	,	PUNCT
ejpam-5048	357	26	0	0	NUM
ejpam-5048	357	27	)	)	PUNCT
ejpam-5048	357	28	.	.	PUNCT
ejpam-5048	358	1	proposition	proposition	NOUN
ejpam-5048	358	2	5	5	NUM
ejpam-5048	358	3	.	.	PUNCT
ejpam-5048	359	1	every	every	DET
ejpam-5048	359	2	closed	close	VERB
ejpam-5048	359	3	qge	qge	NOUN
ejpam-5048	359	4	-	-	NOUN
ejpam-5048	359	5	filter	filter	NOUN
ejpam-5048	359	6	f	f	NOUN
ejpam-5048	359	7	of	of	ADP
ejpam-5048	359	8	x	x	PUNCT
ejpam-5048	359	9	satisfies	satisfie	NOUN
ejpam-5048	359	10	:	:	PUNCT
ejpam-5048	359	11	(	(	PUNCT
ejpam-5048	359	12	∀κ	∀κ	NUM
ejpam-5048	359	13	∈	∈	X
ejpam-5048	359	14	x)(κ	x)(κ	PUNCT
ejpam-5048	360	1	∈	∈	PROPN
ejpam-5048	360	2	f	f	PROPN
ejpam-5048	360	3	⇒	⇒	PROPN
ejpam-5048	360	4	κ	κ	PROPN
ejpam-5048	360	5	∗	∗	NOUN
ejpam-5048	360	6	1	1	NUM
ejpam-5048	360	7	∈	∈	PROPN
ejpam-5048	360	8	f	f	PROPN
ejpam-5048	360	9	)	)	PUNCT
ejpam-5048	360	10	.	.	PUNCT
ejpam-5048	361	1	(	(	PUNCT
ejpam-5048	361	2	28	28	NUM
ejpam-5048	361	3	)	)	PUNCT
ejpam-5048	361	4	proof	proof	NOUN
ejpam-5048	361	5	.	.	PUNCT
ejpam-5048	362	1	it	it	PRON
ejpam-5048	362	2	is	be	AUX
ejpam-5048	362	3	clear	clear	ADJ
ejpam-5048	362	4	.	.	PUNCT
ejpam-5048	363	1	remark	remark	PROPN
ejpam-5048	363	2	4	4	NUM
ejpam-5048	363	3	.	.	PUNCT
ejpam-5048	364	1	the	the	DET
ejpam-5048	364	2	proposition	proposition	NOUN
ejpam-5048	364	3	5	5	NUM
ejpam-5048	364	4	is	be	AUX
ejpam-5048	364	5	not	not	PART
ejpam-5048	364	6	applicable	applicable	ADJ
ejpam-5048	364	7	when	when	SCONJ
ejpam-5048	364	8	the	the	DET
ejpam-5048	364	9	qge	qge	NOUN
ejpam-5048	364	10	-	-	NOUN
ejpam-5048	364	11	filter	filter	NOUN
ejpam-5048	364	12	f	f	NOUN
ejpam-5048	364	13	of	of	ADP
ejpam-5048	364	14	x	x	PROPN
ejpam-5048	364	15	is	be	AUX
ejpam-5048	364	16	not	not	PART
ejpam-5048	364	17	closed	closed	ADJ
ejpam-5048	364	18	.	.	PUNCT
ejpam-5048	365	1	in	in	ADP
ejpam-5048	365	2	fact	fact	NOUN
ejpam-5048	365	3	,	,	PUNCT
ejpam-5048	365	4	the	the	DET
ejpam-5048	365	5	qge	qge	NOUN
ejpam-5048	365	6	-	-	NOUN
ejpam-5048	365	7	filter	filter	NOUN
ejpam-5048	365	8	k	k	NOUN
ejpam-5048	365	9	:	:	PUNCT
ejpam-5048	365	10	=	=	SYM
ejpam-5048	365	11	x	x	SYM
ejpam-5048	365	12	×	×	PROPN
ejpam-5048	365	13	n0	n0	NUM
ejpam-5048	365	14	of	of	ADP
ejpam-5048	365	15	y	y	PRON
ejpam-5048	365	16	which	which	PRON
ejpam-5048	365	17	is	be	AUX
ejpam-5048	365	18	described	describe	VERB
ejpam-5048	365	19	in	in	ADP
ejpam-5048	365	20	example	example	NOUN
ejpam-5048	365	21	14	14	NUM
ejpam-5048	365	22	is	be	AUX
ejpam-5048	365	23	not	not	PART
ejpam-5048	365	24	closed	close	VERB
ejpam-5048	365	25	(	(	PUNCT
ejpam-5048	365	26	see	see	VERB
ejpam-5048	365	27	example	example	NOUN
ejpam-5048	365	28	24	24	NUM
ejpam-5048	365	29	)	)	PUNCT
ejpam-5048	365	30	,	,	PUNCT
ejpam-5048	365	31	and	and	CCONJ
ejpam-5048	365	32	(	(	PUNCT
ejpam-5048	365	33	κ	κ	NOUN
ejpam-5048	365	34	,	,	PUNCT
ejpam-5048	365	35	5	5	NUM
ejpam-5048	365	36	)	)	PUNCT
ejpam-5048	365	37	∈	∈	PROPN
ejpam-5048	365	38	k	k	PROPN
ejpam-5048	365	39	for	for	ADP
ejpam-5048	365	40	all	all	DET
ejpam-5048	365	41	κ	κ	PART
ejpam-5048	365	42	∈	∈	PROPN
ejpam-5048	365	43	x.	x.	NOUN
ejpam-5048	366	1	but	but	CCONJ
ejpam-5048	366	2	(	(	PUNCT
ejpam-5048	366	3	κ	κ	NOUN
ejpam-5048	366	4	,	,	PUNCT
ejpam-5048	366	5	5	5	NUM
ejpam-5048	366	6	)	)	PUNCT
ejpam-5048	366	7	∗	∗	NOUN
ejpam-5048	366	8	1	1	NUM
ejpam-5048	366	9	=	=	SYM
ejpam-5048	366	10	(	(	PUNCT
ejpam-5048	366	11	κ	κ	NOUN
ejpam-5048	366	12	,	,	PUNCT
ejpam-5048	366	13	5	5	NUM
ejpam-5048	366	14	)	)	PUNCT
ejpam-5048	366	15	∗	∗	NOUN
ejpam-5048	366	16	(	(	PUNCT
ejpam-5048	366	17	1x	1x	NUM
ejpam-5048	366	18	,	,	PUNCT
ejpam-5048	366	19	0	0	NUM
ejpam-5048	366	20	)	)	PUNCT
ejpam-5048	366	21	=	=	SYM
ejpam-5048	366	22	(	(	PUNCT
ejpam-5048	366	23	κ	κ	X
ejpam-5048	366	24	∗x	∗x	VERB
ejpam-5048	366	25	1x	1x	NUM
ejpam-5048	366	26	,	,	PUNCT
ejpam-5048	366	27	0−	0−	NUM
ejpam-5048	366	28	5	5	NUM
ejpam-5048	366	29	)	)	PUNCT
ejpam-5048	366	30	=	=	SYM
ejpam-5048	366	31	(	(	PUNCT
ejpam-5048	366	32	κ	κ	X
ejpam-5048	366	33	∗x	∗x	VERB
ejpam-5048	366	34	1x	1x	NUM
ejpam-5048	366	35	,	,	PUNCT
ejpam-5048	366	36	−5	−5	NOUN
ejpam-5048	366	37	)	)	PUNCT
ejpam-5048	366	38	/∈	/∈	PUNCT
ejpam-5048	367	1	k.	k.	PROPN
ejpam-5048	368	1	we	we	PRON
ejpam-5048	368	2	present	present	VERB
ejpam-5048	368	3	the	the	DET
ejpam-5048	368	4	following	follow	VERB
ejpam-5048	368	5	open	open	ADJ
ejpam-5048	368	6	question	question	NOUN
ejpam-5048	368	7	.	.	PUNCT
ejpam-5048	369	1	question	question	NOUN
ejpam-5048	369	2	8	8	NUM
ejpam-5048	369	3	.	.	PUNCT
ejpam-5048	370	1	if	if	SCONJ
ejpam-5048	370	2	a	a	DET
ejpam-5048	370	3	qge	qge	NOUN
ejpam-5048	370	4	-	-	NOUN
ejpam-5048	370	5	filter	filter	NOUN
ejpam-5048	370	6	f	f	NOUN
ejpam-5048	370	7	of	of	ADP
ejpam-5048	370	8	x	x	PRON
ejpam-5048	370	9	satisfies	satisfy	VERB
ejpam-5048	370	10	the	the	DET
ejpam-5048	370	11	condition	condition	NOUN
ejpam-5048	370	12	(	(	PUNCT
ejpam-5048	370	13	28	28	NUM
ejpam-5048	370	14	)	)	PUNCT
ejpam-5048	370	15	,	,	PUNCT
ejpam-5048	370	16	then	then	ADV
ejpam-5048	370	17	is	be	AUX
ejpam-5048	370	18	it	it	PRON
ejpam-5048	370	19	closed	closed	ADJ
ejpam-5048	370	20	?	?	PUNCT
ejpam-5048	371	1	theorem	theorem	VERB
ejpam-5048	371	2	11	11	NUM
ejpam-5048	371	3	.	.	PUNCT
ejpam-5048	372	1	the	the	DET
ejpam-5048	372	2	intersection	intersection	NOUN
ejpam-5048	372	3	of	of	ADP
ejpam-5048	372	4	two	two	NUM
ejpam-5048	372	5	qge	qge	NOUN
ejpam-5048	372	6	-	-	PUNCT
ejpam-5048	372	7	filters	filter	NOUN
ejpam-5048	372	8	is	be	AUX
ejpam-5048	372	9	a	a	DET
ejpam-5048	372	10	qge	qge	NOUN
ejpam-5048	372	11	-	-	NOUN
ejpam-5048	372	12	filter	filter	NOUN
ejpam-5048	372	13	.	.	PUNCT
ejpam-5048	373	1	proof	proof	NOUN
ejpam-5048	373	2	.	.	PUNCT
ejpam-5048	374	1	this	this	PRON
ejpam-5048	374	2	can	can	AUX
ejpam-5048	374	3	be	be	AUX
ejpam-5048	374	4	easily	easily	ADV
ejpam-5048	374	5	checked	check	VERB
ejpam-5048	374	6	.	.	PUNCT
ejpam-5048	375	1	the	the	DET
ejpam-5048	375	2	union	union	NOUN
ejpam-5048	375	3	of	of	ADP
ejpam-5048	375	4	two	two	NUM
ejpam-5048	375	5	qge	qge	NOUN
ejpam-5048	375	6	-	-	NOUN
ejpam-5048	375	7	filters	filter	NOUN
ejpam-5048	375	8	may	may	AUX
ejpam-5048	375	9	not	not	PART
ejpam-5048	375	10	be	be	AUX
ejpam-5048	375	11	a	a	DET
ejpam-5048	375	12	qge	qge	NOUN
ejpam-5048	375	13	-	-	NOUN
ejpam-5048	375	14	filter	filter	NOUN
ejpam-5048	375	15	as	as	SCONJ
ejpam-5048	375	16	shown	show	VERB
ejpam-5048	375	17	in	in	ADP
ejpam-5048	375	18	the	the	DET
ejpam-5048	375	19	following	follow	VERB
ejpam-5048	375	20	example	example	NOUN
ejpam-5048	375	21	.	.	PUNCT
ejpam-5048	376	1	example	example	NOUN
ejpam-5048	376	2	27	27	NUM
ejpam-5048	376	3	.	.	PUNCT
ejpam-5048	377	1	consider	consider	VERB
ejpam-5048	377	2	the	the	DET
ejpam-5048	377	3	qge	qge	NOUN
ejpam-5048	377	4	-	-	NOUN
ejpam-5048	377	5	algebra	algebra	NOUN
ejpam-5048	377	6	x	x	PUNCT
ejpam-5048	377	7	given	give	VERB
ejpam-5048	377	8	in	in	ADP
ejpam-5048	377	9	example	example	NOUN
ejpam-5048	377	10	2	2	X
ejpam-5048	377	11	.	.	PUNCT
ejpam-5048	378	1	it	it	PRON
ejpam-5048	378	2	is	be	AUX
ejpam-5048	378	3	routine	routine	ADJ
ejpam-5048	378	4	to	to	PART
ejpam-5048	378	5	verify	verify	VERB
ejpam-5048	378	6	that	that	SCONJ
ejpam-5048	378	7	the	the	DET
ejpam-5048	378	8	set	set	NOUN
ejpam-5048	378	9	e1	e1	NOUN
ejpam-5048	378	10	=	=	SYM
ejpam-5048	378	11	{	{	PUNCT
ejpam-5048	378	12	1	1	NUM
ejpam-5048	378	13	,	,	PUNCT
ejpam-5048	378	14	a	a	PRON
ejpam-5048	378	15	}	}	PUNCT
ejpam-5048	378	16	and	and	CCONJ
ejpam-5048	378	17	e2	e2	PROPN
ejpam-5048	378	18	=	=	PUNCT
ejpam-5048	378	19	{	{	PUNCT
ejpam-5048	378	20	1	1	NUM
ejpam-5048	378	21	,	,	PUNCT
ejpam-5048	378	22	c	c	NOUN
ejpam-5048	378	23	}	}	PUNCT
ejpam-5048	378	24	are	be	AUX
ejpam-5048	378	25	qge	qge	NOUN
ejpam-5048	378	26	-	-	NOUN
ejpam-5048	378	27	filters	filter	NOUN
ejpam-5048	378	28	of	of	ADP
ejpam-5048	378	29	x.	x.	NOUN
ejpam-5048	378	30	but	but	CCONJ
ejpam-5048	378	31	e1	e1	PROPN
ejpam-5048	378	32	∪	∪	PROPN
ejpam-5048	378	33	e2	e2	PROPN
ejpam-5048	378	34	=	=	PUNCT
ejpam-5048	378	35	{	{	PUNCT
ejpam-5048	378	36	1	1	NUM
ejpam-5048	378	37	,	,	PUNCT
ejpam-5048	378	38	a	a	DET
ejpam-5048	378	39	,	,	PUNCT
ejpam-5048	378	40	c	c	NOUN
ejpam-5048	378	41	}	}	PUNCT
ejpam-5048	378	42	is	be	AUX
ejpam-5048	378	43	not	not	PART
ejpam-5048	378	44	a	a	DET
ejpam-5048	378	45	qge	qge	NOUN
ejpam-5048	378	46	-	-	NOUN
ejpam-5048	378	47	filter	filter	NOUN
ejpam-5048	378	48	of	of	ADP
ejpam-5048	378	49	x	x	PRON
ejpam-5048	378	50	since	since	SCONJ
ejpam-5048	378	51	a	a	DET
ejpam-5048	378	52	∈	∈	PROPN
ejpam-5048	378	53	e1	e1	NOUN
ejpam-5048	378	54	∪	∪	NOUN
ejpam-5048	378	55	e2	e2	PROPN
ejpam-5048	378	56	and	and	CCONJ
ejpam-5048	378	57	a	a	DET
ejpam-5048	378	58	∗	∗	NOUN
ejpam-5048	378	59	b	b	NOUN
ejpam-5048	378	60	=	=	SYM
ejpam-5048	378	61	c	c	PROPN
ejpam-5048	378	62	∈	∈	PROPN
ejpam-5048	378	63	e1	e1	PROPN
ejpam-5048	378	64	∪	∪	NOUN
ejpam-5048	378	65	e2	e2	PROPN
ejpam-5048	378	66	but	but	CCONJ
ejpam-5048	378	67	b	b	PROPN
ejpam-5048	378	68	/∈	/∈	PUNCT
ejpam-5048	378	69	e1	e1	PROPN
ejpam-5048	378	70	∪	∪	PROPN
ejpam-5048	378	71	e2	e2	PROPN
ejpam-5048	378	72	.	.	PUNCT
ejpam-5048	379	1	6	6	NUM
ejpam-5048	379	2	.	.	PUNCT
ejpam-5048	379	3	conclusions	conclusion	NOUN
ejpam-5048	379	4	we	we	PRON
ejpam-5048	379	5	have	have	AUX
ejpam-5048	379	6	introduced	introduce	VERB
ejpam-5048	379	7	a	a	DET
ejpam-5048	379	8	new	new	ADJ
ejpam-5048	379	9	type	type	NOUN
ejpam-5048	379	10	of	of	ADP
ejpam-5048	379	11	algebraic	algebraic	ADJ
ejpam-5048	379	12	structure	structure	NOUN
ejpam-5048	379	13	,	,	PUNCT
ejpam-5048	379	14	called	call	VERB
ejpam-5048	379	15	a	a	DET
ejpam-5048	379	16	quasi	quasi	ADJ
ejpam-5048	379	17	ge	ge	PROPN
ejpam-5048	379	18	-	-	PROPN
ejpam-5048	379	19	algebra	algebra	PROPN
ejpam-5048	379	20	(	(	PUNCT
ejpam-5048	379	21	briefly	briefly	ADV
ejpam-5048	379	22	,	,	PUNCT
ejpam-5048	379	23	qge	qge	NOUN
ejpam-5048	379	24	-	-	NOUN
ejpam-5048	379	25	algebra	algebra	NOUN
ejpam-5048	379	26	)	)	PUNCT
ejpam-5048	379	27	and	and	CCONJ
ejpam-5048	379	28	investigated	investigate	VERB
ejpam-5048	379	29	its	its	PRON
ejpam-5048	379	30	properties	property	NOUN
ejpam-5048	379	31	.	.	PUNCT
ejpam-5048	380	1	we	we	PRON
ejpam-5048	380	2	have	have	AUX
ejpam-5048	380	3	introduced	introduce	VERB
ejpam-5048	380	4	the	the	DET
ejpam-5048	380	5	concepts	concept	NOUN
ejpam-5048	380	6	of	of	ADP
ejpam-5048	380	7	qge	qge	NOUN
ejpam-5048	380	8	-	-	PUNCT
ejpam-5048	380	9	subalgebra	subalgebra	NOUN
ejpam-5048	380	10	,	,	PUNCT
ejpam-5048	380	11	qge	qge	NOUN
ejpam-5048	380	12	-	-	NOUN
ejpam-5048	380	13	filter	filter	NOUN
ejpam-5048	380	14	,	,	PUNCT
ejpam-5048	380	15	closed	close	VERB
ejpam-5048	380	16	qge	qge	NOUN
ejpam-5048	380	17	-	-	NOUN
ejpam-5048	380	18	filter	filter	NOUN
ejpam-5048	380	19	and	and	CCONJ
ejpam-5048	380	20	strong	strong	ADJ
ejpam-5048	380	21	qge	qge	NOUN
ejpam-5048	380	22	-	-	NOUN
ejpam-5048	380	23	filter	filter	NOUN
ejpam-5048	380	24	of	of	ADP
ejpam-5048	380	25	a	a	DET
ejpam-5048	380	26	qge	qge	NOUN
ejpam-5048	380	27	-	-	NOUN
ejpam-5048	380	28	algebra	algebra	NOUN
ejpam-5048	380	29	and	and	CCONJ
ejpam-5048	380	30	discussed	discuss	VERB
ejpam-5048	380	31	their	their	PRON
ejpam-5048	380	32	relationships	relationship	NOUN
ejpam-5048	380	33	between	between	ADP
ejpam-5048	380	34	them	they	PRON
ejpam-5048	380	35	.	.	PUNCT
ejpam-5048	381	1	we	we	PRON
ejpam-5048	381	2	have	have	AUX
ejpam-5048	381	3	provided	provide	VERB
ejpam-5048	381	4	conditions	condition	NOUN
ejpam-5048	381	5	for	for	ADP
ejpam-5048	381	6	a	a	DET
ejpam-5048	381	7	subset	subset	NOUN
ejpam-5048	381	8	of	of	ADP
ejpam-5048	381	9	qge	qge	NOUN
ejpam-5048	381	10	-	-	NOUN
ejpam-5048	381	11	algebra	algebra	NOUN
ejpam-5048	381	12	to	to	PART
ejpam-5048	381	13	be	be	AUX
ejpam-5048	381	14	a	a	DET
ejpam-5048	381	15	qge	qge	NOUN
ejpam-5048	381	16	-	-	NOUN
ejpam-5048	381	17	filter	filter	NOUN
ejpam-5048	381	18	.	.	PUNCT
ejpam-5048	382	1	in	in	ADP
ejpam-5048	382	2	our	our	PRON
ejpam-5048	382	3	future	future	ADJ
ejpam-5048	382	4	work	work	NOUN
ejpam-5048	382	5	,	,	PUNCT
ejpam-5048	382	6	we	we	PRON
ejpam-5048	382	7	will	will	AUX
ejpam-5048	382	8	introduce	introduce	VERB
ejpam-5048	382	9	different	different	ADJ
ejpam-5048	382	10	types	type	NOUN
ejpam-5048	382	11	of	of	ADP
ejpam-5048	382	12	qge	qge	NOUN
ejpam-5048	382	13	-	-	NOUN
ejpam-5048	382	14	filters	filter	NOUN
ejpam-5048	382	15	of	of	ADP
ejpam-5048	382	16	a	a	DET
ejpam-5048	382	17	qge	qge	NOUN
ejpam-5048	382	18	-	-	NOUN
ejpam-5048	382	19	algebra	algebra	NOUN
ejpam-5048	382	20	and	and	CCONJ
ejpam-5048	382	21	investigate	investigate	VERB
ejpam-5048	382	22	their	their	PRON
ejpam-5048	382	23	properties	property	NOUN
ejpam-5048	382	24	.	.	PUNCT
ejpam-5048	383	1	conflicts	conflict	NOUN
ejpam-5048	383	2	of	of	ADP
ejpam-5048	383	3	interest	interest	NOUN
ejpam-5048	383	4	or	or	CCONJ
ejpam-5048	383	5	competing	compete	VERB
ejpam-5048	383	6	interests	interest	NOUN
ejpam-5048	383	7	the	the	DET
ejpam-5048	383	8	authors	author	NOUN
ejpam-5048	383	9	declare	declare	VERB
ejpam-5048	383	10	that	that	SCONJ
ejpam-5048	383	11	they	they	PRON
ejpam-5048	383	12	have	have	VERB
ejpam-5048	383	13	no	no	DET
ejpam-5048	383	14	conflicts	conflict	NOUN
ejpam-5048	383	15	of	of	ADP
ejpam-5048	383	16	interest	interest	NOUN
ejpam-5048	383	17	.	.	PUNCT
ejpam-5048	384	1	data	datum	NOUN
ejpam-5048	384	2	and	and	CCONJ
ejpam-5048	384	3	code	code	NOUN
ejpam-5048	384	4	availability	availability	NOUN
ejpam-5048	384	5	no	no	DET
ejpam-5048	384	6	data	datum	NOUN
ejpam-5048	384	7	were	be	AUX
ejpam-5048	384	8	used	use	VERB
ejpam-5048	384	9	to	to	PART
ejpam-5048	384	10	support	support	VERB
ejpam-5048	384	11	this	this	DET
ejpam-5048	384	12	study	study	NOUN
ejpam-5048	384	13	references	reference	VERB
ejpam-5048	384	14	581	581	NUM
ejpam-5048	384	15	supplementary	supplementary	ADJ
ejpam-5048	384	16	information	information	NOUN
ejpam-5048	384	17	not	not	PART
ejpam-5048	384	18	applicable	applicable	ADJ
ejpam-5048	384	19	ethical	ethical	ADJ
ejpam-5048	384	20	approval	approval	NOUN
ejpam-5048	384	21	this	this	DET
ejpam-5048	384	22	article	article	NOUN
ejpam-5048	384	23	does	do	AUX
ejpam-5048	384	24	not	not	PART
ejpam-5048	384	25	contain	contain	VERB
ejpam-5048	384	26	any	any	DET
ejpam-5048	384	27	studies	study	NOUN
ejpam-5048	384	28	with	with	ADP
ejpam-5048	384	29	human	human	ADJ
ejpam-5048	384	30	participants	participant	NOUN
ejpam-5048	384	31	or	or	CCONJ
ejpam-5048	384	32	animals	animal	NOUN
ejpam-5048	384	33	performed	perform	VERB
ejpam-5048	384	34	by	by	ADP
ejpam-5048	384	35	any	any	PRON
ejpam-5048	384	36	of	of	ADP
ejpam-5048	384	37	the	the	DET
ejpam-5048	384	38	authors	author	NOUN
ejpam-5048	384	39	informed	inform	VERB
ejpam-5048	384	40	consent	consent	VERB
ejpam-5048	384	41	the	the	DET
ejpam-5048	384	42	authors	author	NOUN
ejpam-5048	384	43	are	be	AUX
ejpam-5048	384	44	fully	fully	ADV
ejpam-5048	384	45	aware	aware	ADJ
ejpam-5048	384	46	and	and	CCONJ
ejpam-5048	384	47	satisfied	satisfied	ADJ
ejpam-5048	384	48	with	with	ADP
ejpam-5048	384	49	the	the	DET
ejpam-5048	384	50	contents	content	NOUN
ejpam-5048	384	51	of	of	ADP
ejpam-5048	384	52	the	the	DET
ejpam-5048	384	53	article	article	NOUN
ejpam-5048	384	54	.	.	PUNCT
ejpam-5048	385	1	acknowledgements	acknowledgement	NOUN
ejpam-5048	385	2	the	the	DET
ejpam-5048	385	3	authors	author	NOUN
ejpam-5048	385	4	wish	wish	VERB
ejpam-5048	385	5	to	to	PART
ejpam-5048	385	6	thank	thank	VERB
ejpam-5048	385	7	the	the	DET
ejpam-5048	385	8	anonymous	anonymous	ADJ
ejpam-5048	385	9	reviewers	reviewer	NOUN
ejpam-5048	385	10	for	for	ADP
ejpam-5048	385	11	their	their	PRON
ejpam-5048	385	12	valuable	valuable	ADJ
ejpam-5048	385	13	suggestions	suggestion	NOUN
ejpam-5048	385	14	.	.	PUNCT
ejpam-5048	386	1	references	reference	NOUN
ejpam-5048	386	2	[	[	X
ejpam-5048	386	3	1	1	NUM
ejpam-5048	386	4	]	]	X
ejpam-5048	386	5	r	r	NOUN
ejpam-5048	386	6	k	k	PROPN
ejpam-5048	386	7	bandaru	bandaru	PROPN
ejpam-5048	386	8	,	,	PUNCT
ejpam-5048	386	9	a	a	DET
ejpam-5048	386	10	borumand	borumand	NOUN
ejpam-5048	386	11	saeid	saeid	PROPN
ejpam-5048	386	12	,	,	PUNCT
ejpam-5048	386	13	and	and	CCONJ
ejpam-5048	386	14	y	y	PROPN
ejpam-5048	386	15	b	b	PROPN
ejpam-5048	386	16	jun	jun	PROPN
ejpam-5048	386	17	.	.	PROPN
ejpam-5048	386	18	on	on	ADP
ejpam-5048	386	19	ge	ge	PROPN
ejpam-5048	386	20	-	-	PUNCT
ejpam-5048	386	21	algebras	algebras	PROPN
ejpam-5048	386	22	.	.	PUNCT
ejpam-5048	387	1	bulletin	bulletin	NOUN
ejpam-5048	387	2	of	of	ADP
ejpam-5048	387	3	the	the	DET
ejpam-5048	387	4	section	section	NOUN
ejpam-5048	387	5	of	of	ADP
ejpam-5048	387	6	logic	logic	NOUN
ejpam-5048	387	7	,	,	PUNCT
ejpam-5048	387	8	50(1):81–96	50(1):81–96	NUM
ejpam-5048	387	9	,	,	PUNCT
ejpam-5048	387	10	2021	2021	NUM
ejpam-5048	387	11	.	.	PUNCT
ejpam-5048	388	1	[	[	X
ejpam-5048	388	2	2	2	NUM
ejpam-5048	388	3	]	]	PUNCT
ejpam-5048	388	4	r	r	NOUN
ejpam-5048	388	5	k	k	PROPN
ejpam-5048	388	6	bandaru	bandaru	PROPN
ejpam-5048	388	7	,	,	PUNCT
ejpam-5048	388	8	a	a	DET
ejpam-5048	388	9	borumand	borumand	NOUN
ejpam-5048	388	10	saeid	saeid	PROPN
ejpam-5048	388	11	,	,	PUNCT
ejpam-5048	388	12	and	and	CCONJ
ejpam-5048	388	13	y	y	PROPN
ejpam-5048	388	14	b	b	PROPN
ejpam-5048	388	15	jun	jun	PROPN
ejpam-5048	388	16	.	.	PROPN
ejpam-5048	388	17	belligerent	belligerent	ADJ
ejpam-5048	388	18	ge	ge	PROPN
ejpam-5048	388	19	-	-	NOUN
ejpam-5048	388	20	filter	filter	NOUN
ejpam-5048	388	21	in	in	ADP
ejpam-5048	388	22	ge	ge	PROPN
ejpam-5048	388	23	-	-	PUNCT
ejpam-5048	388	24	algebras	algebras	PROPN
ejpam-5048	388	25	.	.	PUNCT
ejpam-5048	389	1	journal	journal	PROPN
ejpam-5048	389	2	of	of	ADP
ejpam-5048	389	3	the	the	DET
ejpam-5048	389	4	indonesian	indonesian	PROPN
ejpam-5048	389	5	mathematical	mathematical	ADJ
ejpam-5048	389	6	society	society	NOUN
ejpam-5048	389	7	,	,	PUNCT
ejpam-5048	389	8	28(1):31–43	28(1):31–43	NUM
ejpam-5048	389	9	,	,	PUNCT
ejpam-5048	389	10	2022	2022	NUM
ejpam-5048	389	11	.	.	PUNCT
ejpam-5048	390	1	[	[	X
ejpam-5048	390	2	3	3	X
ejpam-5048	390	3	]	]	X
ejpam-5048	390	4	a	a	DET
ejpam-5048	390	5	diego	diego	NOUN
ejpam-5048	390	6	.	.	PUNCT
ejpam-5048	391	1	sur	sur	PROPN
ejpam-5048	391	2	les	les	PROPN
ejpam-5048	391	3	algebres	algebres	PROPN
ejpam-5048	391	4	de	de	PROPN
ejpam-5048	391	5	hilbert	hilbert	PROPN
ejpam-5048	391	6	.	.	PUNCT
ejpam-5048	392	1	collection	collection	PROPN
ejpam-5048	392	2	de	de	X
ejpam-5048	392	3	logique	logique	X
ejpam-5048	392	4	mathematique	mathematique	PROPN
ejpam-5048	392	5	.	.	PUNCT
ejpam-5048	393	1	series	series	PROPN
ejpam-5048	393	2	a.	a.	PROPN
ejpam-5048	393	3	,	,	PUNCT
ejpam-5048	393	4	21:1–56	21:1–56	NUM
ejpam-5048	393	5	,	,	PUNCT
ejpam-5048	393	6	1966	1966	NUM
ejpam-5048	393	7	.	.	PUNCT
ejpam-5048	394	1	[	[	X
ejpam-5048	394	2	4	4	NUM
ejpam-5048	394	3	]	]	X
ejpam-5048	394	4	h	h	PROPN
ejpam-5048	394	5	s	s	PROPN
ejpam-5048	394	6	kim	kim	PROPN
ejpam-5048	394	7	and	and	CCONJ
ejpam-5048	394	8	y	y	PROPN
ejpam-5048	394	9	h	h	PROPN
ejpam-5048	394	10	kim	kim	PROPN
ejpam-5048	394	11	.	.	PUNCT
ejpam-5048	395	1	on	on	ADP
ejpam-5048	395	2	be	be	AUX
ejpam-5048	395	3	-	-	PUNCT
ejpam-5048	395	4	algebras	algebra	NOUN
ejpam-5048	395	5	.	.	PUNCT
ejpam-5048	396	1	scientiae	scientiae	PROPN
ejpam-5048	396	2	mathematicae	mathematicae	VERB
ejpam-5048	396	3	japonicae	japonicae	PROPN
ejpam-5048	396	4	online	online	ADV
ejpam-5048	396	5	,	,	PUNCT
ejpam-5048	396	6	e-2006:1299–1302	e-2006:1299–1302	NOUN
ejpam-5048	396	7	,	,	PUNCT
ejpam-5048	396	8	2006	2006	NUM
ejpam-5048	396	9	.	.	PUNCT
ejpam-5048	397	1	[	[	X
ejpam-5048	397	2	5	5	NUM
ejpam-5048	397	3	]	]	PUNCT
ejpam-5048	397	4	a	a	DET
ejpam-5048	397	5	rezaei	rezaei	NOUN
ejpam-5048	397	6	,	,	PUNCT
ejpam-5048	397	7	a	a	DET
ejpam-5048	397	8	borumand	borumand	ADJ
ejpam-5048	397	9	saeid	saeid	PROPN
ejpam-5048	397	10	,	,	PUNCT
ejpam-5048	397	11	and	and	CCONJ
ejpam-5048	397	12	r	r	X
ejpam-5048	397	13	a	a	DET
ejpam-5048	397	14	borzooei	borzooei	NOUN
ejpam-5048	397	15	.	.	PUNCT
ejpam-5048	398	1	relation	relation	NOUN
ejpam-5048	398	2	between	between	ADP
ejpam-5048	398	3	hilbert	hilbert	PROPN
ejpam-5048	398	4	algebras	algebras	PROPN
ejpam-5048	398	5	and	and	CCONJ
ejpam-5048	398	6	be	be	AUX
ejpam-5048	398	7	-	-	PUNCT
ejpam-5048	398	8	algebras	algebra	NOUN
ejpam-5048	398	9	.	.	PUNCT
ejpam-5048	399	1	applications	application	NOUN
ejpam-5048	399	2	and	and	CCONJ
ejpam-5048	399	3	applied	apply	VERB
ejpam-5048	399	4	mathematics	mathematic	NOUN
ejpam-5048	399	5	,	,	PUNCT
ejpam-5048	399	6	8(2):573–584	8(2):573–584	NUM
ejpam-5048	399	7	,	,	PUNCT
ejpam-5048	399	8	2013	2013	NUM
ejpam-5048	399	9	.	.	PUNCT